From Förberedande kurs i matematik 2

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		+	{| border="0" cellspacing="0" cellpadding="0" height="30" width="100%"
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		+	{{Selected tab|[[1.3 Maximum and minimum problems|Theory]]}}
		+	{{Not selected tab|[[1.3 Exercises|Exercises]]}}
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	{{Info|		{{Info|
-	'''Innehåll:'''	+	'''Contents:'''
-	* Kurvskissering	+	* Sketching curves
-	* Max- och minproblem	+	* Maximum and minimum problems
	}}		}}
	{{Info|		{{Info|
-	'''Lärandemål:'''	+	'''Learning outcomes:'''
-	Efter detta avsnitt ska du ha lärt dig att:	+	After this section, you will have learned:
-	* Kunna definitionen av strängt växande funktion, strängt avtagande funktion, lokalt maximum, lokalt minimum, globalt maximum, globalt minimum.	+	* The definition of strictly increasing, strictly decreasing, local maximum, local minimum, global maximum, global minimum.
-	* Veta att om <math>f^{\,\prime}>0</math> i ett intervall så är <math>f</math> strängt växande i intervallet, och att om <math>f^{\,\prime}<0</math> i ett intervall så är <math>f</math> strängt avtagande i intervallet.	+	* That if <math>f^{\,\prime}>0</math> in an interval then <math>f</math> is strictly increasing in the interval, and that if <math>f^{\,\prime}<0</math> in an interval then <math>f</math> is strictly decreasing in the interval.
-	* Kunna bestämma lokala max- och minpunkter samt terasspunkter genom teckenstudium av derivatan.	+	* To locate stationary points and, by studying the sign of the derivative, classify them as local maxima, local minima, and stationary points of inflexion.
-	* Kunna skissera funktionsgrafer genom att göra en teckentabell över derivatan.	+	* To sketch the graph of a function by constructing a table of signs for the derivative.
-	* Kunna bestämma globala och lokala max- och minpunkter genom 1)&nbsp;teckenstudium av derivatan, 2)&nbsp;punkter där funktionen inte är deriverbar, 3)&nbsp;ändpunkter till definitionsmängden.	+	* To determine global and local maxima and minima by 1)&nbsp;studying the sign of the derivative, 2)&nbsp;considering points where the function is not differentiable, 3)&nbsp;examining the endpoints of the interval where the function is defined.
-	* Kunna avgöra lokala max- och minpunkter med tecknet på andraderivatan.	+	* To use the sign of the second derivative to distinguish between local maxima and local minima.
	}}		}}
-	== Växande och avtagande ==	+	== Increasing and decreasing ==
-	Begreppen växande och avtagande känns kanske självklara när man pratar om matematiska funktioner; om funktionen är växande så lutar grafen uppåt och den är avtagande så lutar grafen nedåt.	+	Informally, a function is increasing if its graph slopes upwards and decreasing if its graph slopes downwards.
-	De matematiska definitionerna är följande:	+	The formal mathematical definitions are as follows:
-	En funktion är växande i ett intervall om för alla ''x'' inom intervallet gäller att	+	A function is increasing in an interval if for all <math>x_1</math> and <math>x_2</math> within the interval
-	{{Fristående formel||<math>x_1 < x_2\quad\Rightarrow\quad f(x_1) \le f(x_2)\,\mbox{.}</math>}}	+	{{Displayed math||<math>x_1 < x_2\quad\Rightarrow\quad f(x_1) \le f(x_2)\,\mbox{.}</math>}}
-	En funktion är avtagande i ett intervall om för alla ''x'' inom intervallet gäller att	+	A function is decreasing in an interval if for all <math>x_1</math> and <math>x_2</math> within the interval
-	{{Fristående formel||<math>x_1 < x_2\quad\Rightarrow\quad f(x_1) \ge f(x_2)\,\mbox{.}</math>}}	+	{{Displayed math||<math>x_1 < x_2\quad\Rightarrow\quad f(x_1) \ge f(x_2)\,\mbox{.}</math>}}
-	Med vardagligt språk säger alltså definitionen av t.ex. växande funktion att för ett ''x''-värde till höger på ''x''-axeln är funktionsvärdet minst lika stort som för ett ''x''-värde till vänster. Lägg märke till att denna definition innebär att en funktion kan vara konstant i ett intervall och ändå vara växande eller avtagande. En funktion som är konstant i hela det aktuella intervallet är enligt definitionen både växande och avtagande.	+	In everyday language the definition says, for example, that for an increasing function for any ''x''-value to the right on the ''x''-axis, the value of the function is at least as large as it is for any ''x''-value to the left. Please note that this definition means that a function can be constant in a interval and still be considered to be increasing or decreasing. A function that is constant throughout an interval, according to the definition, is both increasing and decreasing.
		+	If one wants to exclude the possibility of a function being constant on an interval, one talks of ''strictly'' increasing and ''strictly'' decreasing functions:
-	Om man vill utesluta möjligheten att en växande/avtagande funktionen är konstant på ett intervall talar man istället om ''strängt'' växande och ''strängt'' avtagande funktioner.	+	A function is ''strictly'' increasing in an interval if for all <math>x_1</math> and <math>x_2</math> within the interval
-	En funktion är ''strängt'' växande i ett intervall om för alla ''x'' inom intervallet gäller att	+	{{Displayed math||<math>x_1 < x_2\quad\Rightarrow\quad f(x_1) < f(x_2)\,\mbox{.}</math>}}
-	{{Fristående formel||<math>x_1 < x_2\quad\Rightarrow\quad f(x_1) < f(x_2)\,\mbox{.}</math>}}	+	A function is ''strictly'' decreasing in an interval if for <math>x_1</math> och <math>x_2</math> within the interval
-	En funktion är ''strängt'' avtagande i ett intervall om för alla ''x'' inom intervallet gäller att	+	{{Displayed math||<math>x_1 < x_2\quad\Rightarrow\quad f(x_1) > f(x_2)\,\mbox{.}</math>}}
-	{{Fristående formel||<math>x_1 < x_2\quad\Rightarrow\quad f(x_1) > f(x_2)\,\mbox{.}</math>}}	+	(A strictly increasing or decreasing function cannot be constant in any part of the interval.)
-		+
-	(En strängt växande/avtagande funktion får alltså inte vara konstant i någon del av intervallet.)	+
	<div class="exempel">		<div class="exempel">
-	'''Exempel 1'''	+	'''Example 1'''
	<ol type="a">		<ol type="a">
-	<li>Funktionen <math>y= f(x)</math> vars graf ges av figuren nedan längst till	+	<li> The function <math>y= f(x)</math> whose graph is given in the chart below on the far left is increasing in the interval <math>0 \le x \le 6</math>.</li>
-	vänster är växande i intervallet <math>0 \le x \le 6</math>.</li>	+	<li> The function <math>y=-x^3\!/4</math> is a strictly decreasing function.</li>
-	<li>Funktionen <math>y=-x^3/4</math> är en strängt avtagande funktion.</li>	+	<li> The function <math>y=x^2</math> is a strictly increasing function for <math>x \ge 0</math>.</li>
-	<li>Funktionen <math>y=x^2</math> är strängt växande för <math>x \ge 0</math>.</li>	+
	</ol>		</ol>
		+
	{| align="center"		{| align="center"
	|-		|-
-	||{{:1.3 - Figur - Grafen till f(x), där f är styckvis linjär och konstant}}	+	||{{:1.3 - Figure - The graph of f(x), where f is piecewise linear and constant}}
	| width="10px" |		| width="10px" |
-	||{{:1.3 - Figur - Grafen f(x) = -x³/4}}	+	||{{:1.3 - Figure - The graph of f(x) = -x³/4}}
	| width="10px"|		| width="10px"|
-	||{{:1.3 - Figur - Grafen till f(x) = x²}}	+	||{{:1.3 - Figure - The graph of f(x) = x²}}
	|-		|-
-	||<small>Grafen till funktionen<br> i uppgift&nbsp;a</small>	+	||<small>Graph of the function in part <br> &nbsp;a</small>
	||		||
-	||<small>Grafen till funktionen<br> ''f''(''x'')&nbsp;=&nbsp;-&nbsp;''x''³/4</small>	+	||<small>Graph of the function<br> ''f''(''x'')&nbsp;=&nbsp;-&nbsp;''x''³/4</small>
	||		||
-	||<small>Grafen till funktionen<br> ''f''(''x'')&nbsp;=&nbsp;x²</small>	+	||<small>Graph of the function<br> ''f''(''x'')&nbsp;=&nbsp;x²</small>
	|}		|}
	</div>		</div>
-	Derivatan kan givetvis användas för att undersöka om en funktion är växande eller avtagande. Vi har att	+	The derivative may of course be used to examine whether a function is increasing or decreasing. We have that
-	{{Fristående formel||<math>\begin{align*} f^{\,\prime}(x) > 0 \quad&\Rightarrow \quad f(x) \text{ är (strängt) växande.}\\ f^{\,\prime}(x) < 0 \quad&\Rightarrow \quad f(x) \text{ är (strängt) avtagande.} \end{align*}</math>}}	+	{{Displayed math||<math>\begin{align*} f^{\,\prime}(x) > 0 \quad&\Rightarrow \quad f(x) \text{ is (strictly) increasing,}\\ f^{\,\prime}(x) < 0 \quad&\Rightarrow \quad f(x) \text{ is (strictly) decreasing.} \end{align*}</math>}}
-	Observera att även enstaka '''punkter''' där <math>f^{\,\prime}(x) = 0</math> kan ingå i ett strängt växande eller avtagande intervall.	+	Note, however, that this works only one way round. It is entirely possible for a function to be strictly increasing or decreasing on some interval, and for there to be a point, or perhaps more than one, within that interval at which the derivative is zero. As long as any points where the derivative is zero are ''isolated'' (that is, provided the derivative isn't zero anywhere ''close'' to such points, but only ''at'' them), then the function can be strictly increasing or decreasing; problems only arise if the gradient is zero, and the function therefore constant, over some interval.
-	== Kritiska punkter ==	+	== Stationary points ==
-	Punkter där <math>f^{\,\prime}(x) = 0</math> kallas kritiska (eller stationära) punkter och är vanligtvis av tre olika slag:	+	Points where <math>f^{\,\prime}(x) = 0</math> are known as stationary points (the term "critical points" is also sometimes used). They are usually one of three kinds:
-	* lokal maximipunkt med <math>f^{\,\prime}(x) > 0</math> till vänster, och <math>f^{\,\prime}(x) < 0</math> till höger om punkten.	+	* Local maximum with <math>f^{\,\prime}(x) > 0</math> to the left, and <math>f^{\,\prime}(x) < 0</math> to the right of the point.
-	* lokal minimipunkt med <math>f^{\,\prime}(x) < 0</math> till vänster, och <math>f^{\,\prime}(x) > 0</math> till höger om punkten.	+	* Local minimum with <math>f^{\,\prime}(x) < 0</math> to the left, and <math>f^{\,\prime}(x) > 0</math> to the right of the point.
-	* terrasspunkt med <math>f^{\,\prime}(x) < 0</math> eller <math>f^{\,\prime}(x) > 0</math> på båda sidor om punkten.	+	* Stationary point of inflexion with <math>f^{\,\prime}(x) < 0</math> or <math>f^{\,\prime}(x) > 0</math> on both sides of the point.
-	Observera att en punkt kan vara en lokal maximi-/minimipunkt utan att <math>f^{\,\prime}(x) = 0</math>; läs mer i avsnittet om ''[[#Max- och minpunkter (extrempunkter)|max- och minpunkter]]''.	+	Note that a point may be a local maximum or minimum without <math>f^{\,\prime}(x) = 0</math>; learn more about this in the section on ''[[#Max- och minpunkter (extrempunkter)|maxima and minima]]''.
-	<center>{{:1.3 - Figur - Grafen till f(x) = x³ - x⁵}}</center>	+	== Points of inflexion ==
		+
		+	A point of inflexion is a point where the direction of curvature of a graph changes: where, having curved upwards, it begins curving downwards, or vice versa. In other words, it is a local maximum or minimum ''for the gradient''.
		+
		+	One way to think about points of inflexion is to imagine driving a car along a road shaped like the curve; a point of inflexion is any point at which your steering wheel is exactly centred and you are, for that instant, steering neither right nor left.
		+
		+	<center>{{:1.3 - Figure - Different types of inflexion points}}</center>
		+	{| width="85%" align="center"
		+	||<small> At a point of inflexion, the curve's direction of curvature changes; the curve on the left has a ''stationary'' point of inflexion in ''x''&nbsp;=&nbsp;0, where the gradient is zero, but as the other two curves show, not all points of inflexion are stationary points.</small>
		+	|}
		+
		+	We will not study points of inflexion in depth in this section, except to note that a point of inflexion need not necessarily also be a stationary point (though the two can coincide, in which case we would often call the point concerned a "stationary point of inflexion").
		+
		+	<center>{{:1.3 - Figure - The graph of f(x) = x³ - x⁵}}</center>
-	Funktionen i figuren ovan har en lokal minimipunkt för <math>x = -2</math>, terrasspunkt för <math>x = 0</math> och lokal maximipunkt för <math>x = 2</math>.	+	The function in the above figure has a local minimum at <math>x = -2</math>, a stationary point of inflexion at <math>x = 0</math> and a local maximum for <math>x = 2</math>.
-	== Teckentabell ==	+	== Table of signs ==
-	Genom att studera derivatans tecken (+, – eller 0) kan vi alltså få en bra uppfattning om kurvans utseende.	+	By studying the derivative sign (+, - or 0), we can therefore obtain a good idea of the curve's appearance.
-	Detta utnyttjar man i en s.k. ''teckentabell''. Man bestämmer först de ''x''-värden där <math>f^{\,\prime}(x) =0</math> och beräknar sedan derivatans tecken på båda sidor om dessa. Med hjälp av en eller annan "stödpunkt" på kurvan kan man dessutom utifrån teckentabellen skissera kurvan på ett ofta godtagbart sätt.	+	One creates a so called ''table of signs''. One first determines the ''x''-values where <math>f^{\,\prime}(x) =0</math> (together with any where the gradient is undefined) and then calculates the sign of the derivative on either side of these points. With the help of some other "backup" points on the curve and using the table of signs one usually can obtain a satisfactory sketch of the curve.
	<div class="exempel">		<div class="exempel">
-	'''Exempel 2'''	+	'''Example 2'''
-	Gör en teckentabell över funktionen <math>f(x) = x^3 -12x + 6</math> och skissera därefter funktionens graf.	+	Make a table of signs of the derivative of the function <math>f(x) = x^3 -12x + 6</math> and then sketch the graph of the function.
	<br>		<br>
	<br>		<br>
-	Funktionens derivata ges av	+	The functions derivative is given by
-	{{Fristående formel||<math>	+	{{Displayed math||<math>
	f^{\,\prime}(x) = 3x^2 -12 = 3(x^2-4) = 3(x-2)(x+2).</math>}}		f^{\,\prime}(x) = 3x^2 -12 = 3(x^2-4) = 3(x-2)(x+2).</math>}}
-	Faktorn <math>x-2</math> är negativ till vänster om <math>x=2</math> och positiv till höger om <math>x=2</math>. På samma sätt är faktorn <math>x+2</math> negativ till vänster om <math>x=-2</math> och positiv till höger om <math>x=-2</math>. Denna information kan vi också sammanfatta i en tabell:	+	The factor <math>x-2</math> is negative to the left of <math>x=2</math> and positive to the right of <math>x=2</math>. In the same way the factor <math>x+2</math> is negative to the left of <math>x=-2</math> and positive to the right of <math>x=-2</math>. This information can be summarised in a table:
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-	Eftersom derivatan är produkten av <math>x-2</math> och <math>x+2</math> så kan vi bestämma derivatans tecken utifrån faktorernas tecken och ställa upp en följande tabell över derivatans tecken på tallinjen:	+	Since the derivative is the product of <math>x-2</math> and <math>x+2</math> we thus can determine the derivative sign on the basis of the sign of these factors and create the following table of signs for the derivative on the real-number axis :
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-	I tabellens sista rad har vi skrivit ut pilar som visar om funktionen är strängt växande&nbsp;<math>(\,\nearrow\,\,)</math> eller strängt avtagande&nbsp;<math>(\,\searrow\,\,)</math> i respektive intervall samt funktionens värde i de kritiska punkterna <math>x=-2</math> och <math>x=2</math>.	+	In the table's last line, we have given arrows that indicate whether the function is strictly increasing&nbsp;<math>(\,\nearrow\,\,)</math> or strictly decreasing&nbsp;<math>(\,\searrow\,\,)</math> in each interval as well as the value of the function value at the stationary points <math>x=-2</math> and <math>x=2</math>.
-	Från diagrammet ser vi att <math>f(x)</math> har en lokal maximipunkt i <math>(–2, 22)</math> och en lokal minimipunkt i <math>(2, –10)</math>. Grafen kan nu skissas:	+	From the figure, we see that <math>f(x)</math> has a local maximum at <math>(–2, 22)</math> and a local minimum at <math>(2, –10)</math>. The graph now can be sketched:
-	<center>{{:1.3 - Figur - Grafen till f(x) = x³ - 12x + 6}}</center>	+	<center>{{:1.3 - Figure - The graph of f(x) = x³ - 12x + 6}}</center>
	</div>		</div>
-	== Max- och minpunkter (extrempunkter) ==	+	== Maxima and minima (extrema) ==
-	Punkter där en funktion antar sitt största eller minsta värde i jämförelse med omgivningen kallas för ''lokala maximi-'' eller ''minimipunkter'' (förkortas ofta max- och minpunkter). Med ett gemensamt namn kallas dessa punkter för ''extrempunkter''.	+	A point at which a function takes on its largest or smallest value in comparison with its immediate surroundings is called a ''local maximum '' or ''local minimum'' (often abbreviated to max and min). Local maxima and minima are together known as ''extrema''.
-	En extrempunkt kan uppträda i tre olika slags punkter:	+	An extremum may occur in one of three ways:
-	:* i en kritisk punkt (där <math>f^{\,\prime}(x)=0\,</math>).	+	:* At a stationary point (where <math>f^{\,\prime}(x)=0\,</math>).
-	:* i en punkt där derivatan inte existerar (s.k. ''singulär punkt'').	+	:* At a point where the derivative does not exist (known as a ''singular point'').
-	:* i en ändpunkt till definitionsmängden.	+	:* At an endpoint to the interval where the function is defined.
	<div class="exempel">		<div class="exempel">
-	'''Exempel 3'''	+	''' Example 3'''
-	Funktionen nedan har fyra extrempunkter: maxpunkter i <math>x=c</math> och <math>x=e</math>, och minpunkter i <math>x=a</math> och <math>x=d</math>.	+	For the function below there are four extrema: maximum at <math>x=c</math> and <math>x=e</math>, and minimum at <math>x=a</math> and <math>x=d</math>.
-	<center>{{:1.3 - Figur - Grafen till f(x), där f har extrempunkter i x = a, b, c, d, e}}</center>	+	<center>{{:1.3 - Figure - The graph of f(x), where f has extreme points in x = a, b, c, d, e}}</center>
-	I <math>x=a</math>, <math>x=b</math> och <math>x=d</math> är <math>f^{\,\prime}(x) =0</math>, men det är endast i <math>x=a</math> och <math>x=d</math> som vi har extrempunkter, eftersom <math>x=b</math> är en terrasspunkt.	+	At <math>x=a</math>, <math>x=b</math> and <math>x=d</math> one has <math>f^{\,\prime}(x) =0</math>, but it is only at <math>x=a</math> and <math>x=d</math> that there are extrema, since <math>x=b</math> is a stationary point of inflexion.
-	I <math>x=c</math> är inte derivatan definierad (eftersom det är en spets, eller hörn, på kurvan och lutningen inte går att bestämma). Punkten <math>x=e</math> är en ändpunkt.	+	At <math>x=c</math> the derivative is not defined (as it is a cusp or corner of the curve and it is not possible to determine the slope). The point <math>x=e</math> is an endpoint.
	</div>		</div>
-	När man letar efter extrempunkter hos en funktion gäller det alltså att ta reda på och undersöka alla tänkbara kandidater av punkter. En lämplig arbetsgång är:	+	When one is looking for the extrema of a function one must discover and examine all possible candidates for these points. An appropriate working procedures is:
-	:# Derivera funktionen	+	:# Differentiate the function.
-	:# Kontrollera om det finns några punkter där <math>f^{\,\prime}(x)</math> inte är definierad.	+	:# Check to see if there are any points where <math>f^{\,\prime}(x)</math> is not defined.
-	:# Bestäm alla punkter där <math>f^{\,\prime}(x) = 0</math>.	+	:# Determine all points where <math>f^{\,\prime}(x) = 0</math>.
-	:# Gör en teckentabell för att få fram alla extrempunkter.	+	:# Make a table of signs to locate and classify all of the extrema.
-	:# Beräkna funktionsvärdet i alla extrempunkter, samt i eventuella ändpunkter.	+	:# Calculate the value of the function for all the extrema and at any endpoints.
	<div class="exempel">		<div class="exempel">
-	'''Exempel 4'''	+	''' Example 4'''
-	Bestäm alla extrempunkter på kurvan <math>y=3x^4 +4x^3 - 12x^2 + 12</math>.	+	Determine all the extrema of the curve <math>y=3x^4 +4x^3 - 12x^2 + 12</math>.
	<br>		<br>
	<br>		<br>
-	Funktionens derivata ges av	+	The function's derivative is given by
-	{{Fristående formel||<math>	+	{{Displayed math||<math>
	y' = 12x^3 + 12x^2 - 24x = 12x(x^2+x-2)\,\mbox{.}</math>}}		y' = 12x^3 + 12x^2 - 24x = 12x(x^2+x-2)\,\mbox{.}</math>}}
-	För att bestämma hur derivatans tecken varierar över tallinjen försöker vi faktorisera derivatan så långt som möjligt. Vi har redan lyckats bryta ut faktorn <math>12x</math> och vi kan faktorisera det återstående uttrycket <math>x^2+x-2</math> ytterligare genom att hitta dess nollställen	+	In order to determine how the sign of the derivative varies along the real-number axis, we factorise the derivative as completely as possible. We have already managed to take out the factor <math>12x</math> and we can factorise further the remaining term <math>x^2+x-2</math> by identifying its zeros
-	{{Fristående formel||<math>	+	{{Displayed math||<math>
-	x^2+x-2=0\qquad\Leftrightarrow\qquad x=-2\quad\text{eller}\quad x=1.</math>}}	+	x^2+x-2=0\qquad\Leftrightarrow\qquad x=-2\quad\text{or}\quad x=1.</math>}}
-	Detta betyder att <math>x^2+x-2=(x+2)(x-1)</math> och hela derivatan kan skrivas som	+	This means that <math>x^2+x-2=(x+2)(x-1)</math> and the derivative can be rewritten as
-	{{Fristående formel||<math>y' = 12x(x+2)(x-1)\,\mbox{.}</math>}}	+	{{Displayed math||<math>y' = 12x(x+2)(x-1)\,\mbox{.}</math>}}
-	Det går direkt ur denna formel se att derivatan är noll för <math>x=-2</math>, <math>x=0</math> och <math>x=1</math>. Dessutom kan vi se hur derivatans tecken varierar genom att undersöka tecknet för varje enskild faktor i produkten för olika värden på <math>x</math>	+	It can be seen immediately from this that the derivative is zero for <math>x=-2</math>, <math>x=0</math> and <math>x=1</math>. In addition, we can see how the derivatives sign varies by examining the sign of each individual factor in the product for different values of <math>x</math>.
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-	Derivatan är produkten av dessa faktorer och vi får derivatans tecken genom att multiplicera ihop faktorernas tecken i respektive intervall.	+	The derivative is the product of these factors, and we may obtain the sign of the derivative by multiplying together signs of the factors in each interval.
	{| border="1" cellpadding="5" cellspacing="0" align="center"		{| border="1" cellpadding="5" cellspacing="0" align="center"
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	|}		|}
-	Kurvan har alltså lokala minpunkter i <math>(–2, –20)</math> och <math>(1, 7)</math> samt lokal maxpunkt i <math>(0, 12)</math>.	+	The curve has thus local minima at <math>(–2, –20)</math> and <math>(1, 7)</math> and a local maximum at <math>(0, 12)</math>.
	</div>		</div>
	<div class="exempel">		<div class="exempel">
-	'''Exempel 5'''	+	''' Example 5'''
-	Bestäm alla extrempunkter på kurvan <math>y= x - x^{2/3}</math>.	+	Determine all extrema of the curve <math>y= x - x^{2/3}</math>.
	<br>		<br>
	<br>		<br>
-	Derivatan till funktionen ges av	+	The derivative of the function is given by
-	{{Fristående formel||<math>	+	{{Displayed math||<math>
	y' = 1 - \frac{2}{3} x^{-1/3} = 1- \frac {2}{3}		y' = 1 - \frac{2}{3} x^{-1/3} = 1- \frac {2}{3}
-	\cdot \frac{1}{\sqrt[\scriptstyle 3]{x}}\,\mbox{.}</math>}}	+	\, \frac{1}{\sqrt[\scriptstyle 3]{x}}\,\mbox{.}</math>}}
-	Från detta uttryck ser vi att <math>y'</math> inte är definierad för <math>x = 0</math> (vilket dock <math>y</math> är). Detta betyder att funktionen har en singulär punkt i <math>x=0</math>.	+	From this expression, we see that <math>y'</math> is not defined for <math>x = 0</math> (although which <math>y</math> is defined). This means that the function has a singular point at <math>x=0</math>.
-	De kritiska punkterna till funktionen ges av	+	The stationary points of the function are given by
-	{{Fristående formel||<math>	+	{{Displayed math||<math>
	y'=0 \quad \Leftrightarrow \quad		y'=0 \quad \Leftrightarrow \quad
-	1= \frac {2}{3} \cdot \frac{1}{\sqrt[3]{x}}\quad\Leftrightarrow\quad	+	1= \frac {2}{3} \, \frac{1}{\sqrt[3]{x}}\quad\Leftrightarrow\quad
	\sqrt[3]{x} = \tfrac {2}{3}\quad \Leftrightarrow \quad		\sqrt[3]{x} = \tfrac {2}{3}\quad \Leftrightarrow \quad
	x = \bigl(\tfrac{2}{3}\bigr)^3 = \tfrac{8}{27}\,\mbox{.}</math>}}		x = \bigl(\tfrac{2}{3}\bigr)^3 = \tfrac{8}{27}\,\mbox{.}</math>}}
-	De enda punkter där funktionen eventuellt kan ha extrempunkter är alltså <math>x=0</math> och <math>x=\tfrac{8}{27}</math>. För att avgöra dessa punkters karaktär skriver vi upp en teckentabell:	+	The only points at which the function might have an extremum are thus <math>x=0</math> and <math>x=\tfrac{8}{27}</math>. In order to determine the nature of these points we create a table of signs:
	{| border="1" cellpadding="5" cellspacing="0" align="center"		{| border="1" cellpadding="5" cellspacing="0" align="center"
Line 340:		Line 360:
	|width="50px" align="center"| <math>y'</math>		|width="50px" align="center"| <math>y'</math>
	|width="50px" align="center"| <math>+</math>		|width="50px" align="center"| <math>+</math>
-	|width="50px" align="center"| ej def.	+	|width="50px" align="center"| not def.
	|width="50px" align="center"| <math>-</math>		|width="50px" align="center"| <math>-</math>
	|width="50px" align="center"| <math>0</math>		|width="50px" align="center"| <math>0</math>
Line 353:		Line 373:
	|}		|}
-	Kurvan har alltså en lokal maximipunkt i <math>(0, 0)</math> (en spets) och en lokal minimipunkt i <math>(\tfrac{8}{27},-\tfrac{4}{27})\,</math>.	+	The curve has a local maximum at <math>(0, 0)</math> (a cusp) and a local minimum at <math>(\tfrac{8}{27},-\tfrac{4}{27})\,</math>.
-	<center>{{:1.3 - Figur - Grafen till f(x) = x - x^⅔}}</center>	+	<center>{{:1.3 - Figure - The graph of f(x) = x - x^⅔}}</center>
	</div>		</div>
-	== Absolut min/max ==	+	== Global min / max ==
-	En funktion har ett ''absolut'' (eller ''globalt'') maximum (minimum) i en punkt om funktionsvärdet inte är större (mindre) i någon annan punkt i hela definitionsmängden. Ofta kallar man också detta för funktionens största (minsta) värde.	+	A function has a''global'' maximum at a point if its value there is greater than, or at least equal to, its value at any other point where it is defined; similarly, a global minimum is a point where the function's value is less than, or at most equal to, its value anywhere else.
-	För att bestämma en funktions absoluta max. eller min. så måste man alltså hitta alla extrempunkter och beräkna funktionsvärdena i dessa. Om definitionsmängden har ändpunkter måste man givetvis också undersöka funktionens värde i dessa punkter.	+	To determine a function's global max or min one must therefore find all the extrema and calculate the values of the function at them. If the function is defined on an interval with endpoints, one must of course also examine its value at these points.
-	Observera att en funktion kan sakna såväl absolut max. som absolut min. Notera också att en funktion kan ha flera lokala extrempunkter utan att ha ett globalt max. eller min.	+	Note that a function need not have a global max or a global min, even if it has several local extrema.
	<div class="exempel">		<div class="exempel">
-	'''Exempel 6'''	+	''' Example 6'''
-	<center>{{:1.3 - Figur - Två funktioner som saknar min och max resp. min}}</center>	+	<center>{{:1.3 - Figure - Two functions where one has no min nor max, and one has no min}}</center>
-	I den första figuren saknar funktionen såväl globalt maximum som globalt minimum. I den andra figuren saknar funktionen globalt minimum.	+	In the first figure the function has no global maximum nor global minimum. In the second figure the function has no global minimum.
	</div>		</div>
-	I tillämpningar ger omständigheterna ofta en begränsad definitionsmängd, dvs. man betraktar endast en del av funktionens graf. Man måste därför vara vaksam på att ett globalt max. eller min. mycket väl kan ligga i intervallets ändpunkter.	+	In applications, circumstances often dictate that a function has a limited interval where it is defined, i.e. one only studies part of the graph of the function. One must therefore be careful in case the global max or min is at an endpoint of the interval.
-	<center>{{:1.3 - Figur - Funktion med lokala och globala extrempunkter}}</center>	+	<center>{{:1.3 - Figure - A function with local and global extreme points}}</center>
-	Funktionen ovan betraktas endast i intervallet <math>a\le x \le e</math>. Vi ser att funktionens minsta värde i detta intervall inträffar i den kritiska punkten <math>x=b</math>, medan största värdet återfinns i ändpunkten <math>x=e</math>.	+	The above function is only of interest in the interval <math>a\le x \le e</math>. We see that the minimum value of the function in this interval occurs at the stationary point <math>x=b</math>, while the maximim value is found at the endpoint <math>x=e</math>.
	<div class="exempel">		<div class="exempel">
-	'''Exempel 7'''	+	''' Example 7'''
-	Bestäm största och minsta värde för funktionen <math>f(x) = x^3 -3x + 2</math> i intervallet <math>-0{,}5 \le x \le 1\,</math>.	+	Determine the maximum and minimum value of the function <math>f(x) = x^3 -3x + 2</math> in the interval <math>-0\textrm{.}5 \le x \le 1\,</math>.
	<br>		<br>
	<br>		<br>
-	Vi deriverar funktionen <math>f^{\,\prime}(x) = 3x^2 -3</math> och sätter derivatan lika med noll för att få fram alla kritiska punkter	+	We differentiate the function, <math>f^{\,\prime}(x) = 3x^2 -3</math>, and put the derivative equal to zero to obtain all the stationary points
-	{{Fristående formel||<math>f^{\,\prime}(x) = 0 \quad \Leftrightarrow \quad x^2 = 1 \quad \Leftrightarrow \quad x= \pm 1\,\mbox{.}</math>}}	+	{{Displayed math||<math>f^{\,\prime}(x) = 0 \quad \Leftrightarrow \quad x^2 = 1 \quad \Leftrightarrow \quad x= \pm 1\,\mbox{.}</math>}}
-	Punkten <math>x = –1</math> ligger dock utanför den aktuella definitionsmängden och <math>x = 1</math> sammanfaller med definitionsmängdens ena ändpunkt. Eftersom funktionen saknar singulära punkten (funktionen är deriverbar överallt) måste funktionens största och minsta värde antas i intervallets ändpunkter.	+	The point <math>x = –1</math> is outside the interval on which the function is defined, and <math>x = 1</math> lies at one endpoint of this interval. Since the function has no singular points (it is differentiable everywhere), its maximum and minimum must be at the interval's endpoints,
-	{{Fristående formel||<math>\begin{align*} f(-0{,}5) &= 3{,}375\\[4pt] f(1)&=0 \end{align*}</math>}}	+	{{Displayed math||<math>\begin{align*} f(-0\textrm{.}5) &= 3\textrm{.}375\,\mbox{,}\\[4pt] f(1)&=0\,\mbox{.} \end{align*}</math>}}
-	Funktionens största värde i det givna intervallet är alltså <math>3{,}375</math>. Minsta värdet är <math>0</math> (se figuren).	+	The function's maximum value on the given interval is thus <math>3\textrm{.}375</math>. The minimum value is <math>0</math> (see the figure).
-	<center>{{:1.3 - Figur - Grafen till f(x) = x³ - 3x + 2}}</center>	+	<center>{{:1.3 - Figure - The graph of f(x) = x³ - 3x + 2}}</center>
-	Figuren visar funktionens hela graf streckad, med den del som ligger inom det givna intervallet heldragen.	+	The figure shows the function with the whole graph as a dashed curve , with the part that is within the given interval appearing as a continuous curve.
	</div>		</div>
-	== Andraderivatan ==	+	== The second derivative ==
-	Tecknet på derivatan av en funktion ger oss information om huruvida funktionen är växande eller avtagande. På samma sätt kan andraderivatans tecken visa om förstaderivatan är växande eller avtagande. Detta kan man bl.a. utnyttja för att ta reda på om en given extrempunkt är en max-, eller minpunkt.	+	The sign of the derivative of a function gives us information about whether the function is increasing or decreasing. Similarly, the sign of the ''second'' derivative can show if the first order derivative is increasing or decreasing. This can , among other things, be used to find out whether a given extremum is a maximum or minimum.
-	Om funktionen <math>f(x)</math> har en kritisk punkt i <math>x=a</math> där <math>f^{\,\prime\prime}(a)<0</math>, då gäller att	+	If the function <math>f(x)</math> has a stationary point at <math>x=a</math> where <math>f^{\,\prime\prime}(a)<0</math>, then
-	# Derivatan <math>f^{\,\prime}(x)</math> är strängt avtagande i en omgivning kring <math>x=a</math>.	+	# The derivative <math>f^{\,\prime}(x)</math> is strictly decreasing in some interval surrounding <math>x=a</math>.
-	# Eftersom <math>f^{\,\prime}(a)=0</math> är alltså <math>f^{\,\prime}(x)>0</math> till vänster om <math>x=a</math> och <math>f^{\,\prime}(x)<0</math> till höger om <math>x=a</math>.	+	# Since <math>f^{\,\prime}(a)=0</math> then <math>f^{\,\prime}(x)>0</math> to the left of <math>x=a</math> and <math>f^{\,\prime}(x)<0</math> to the right of <math>x=a</math>.
-	# Detta medför att funktionen <math>f(x)</math> har en lokal maximipunkt i <math>x=a</math>.	+	# This means that the function <math>f(x)</math> has a local maximum at <math>x=a</math>.
-	<center>{{:1.3 - Figur - Tangenten till funktion med negativ andraderivata}}</center>	+	<center>{{:1.3 - Figure - The tangent of a function with negative second order derivative}}</center>
	{| width="80%" align="center"		{| width="80%" align="center"
-	||<small>Om derivatan är positiv till vänster om ''x''&nbsp;=&nbsp;''a'' och negativ till höger om ''x''&nbsp;=&nbsp;''a'' så har funktionen ett lokalt maximum i ''x''&nbsp;=&nbsp;''a''.</small>	+	||<small> If the derivative is positive to the left of ''x''&nbsp;=&nbsp;''a'' and negative to the right of ''x''&nbsp;=&nbsp;''a'' the function has a local maximum at ''x''&nbsp;=&nbsp;''a''.</small>
	|}		|}
-	Om funktionen <math>f(x)</math> har en kritisk punkt i <math>x=a</math> där <math>f^{\,\prime\prime}(a)>0</math>, då gäller att	+	If the function <math>f(x)</math> has a stationary point at <math>x=a</math> where <math>f^{\,\prime\prime}(a)>0</math>, then
-	# Derivatan <math>f^{\,\prime}(x)</math> är strängt växande i en omgivning kring <math>x=a</math>.	+	# The derivative <math>f^{\,\prime}(x)</math> is strictly increasing in some interval around <math>x=a</math>.
-	# Eftersom <math>f^{\,\prime}(a)=0</math> är alltså <math>f^{\,\prime}(x)<0</math> till vänster om <math>x=a</math> och <math>f^{\,\prime}(x)>0</math> till höger om <math>x=a</math>.	+	# Since <math>f^{\,\prime}(a)=0</math> then <math>f^{\,\prime}(x)<0</math> to the left of <math>x=a</math> and <math>f^{\,\prime}(x)>0</math> to the right of <math>x=a</math>.
-	# Detta medför att funktionen <math>f(x)</math> har en lokal minimipunkt i <math>x=a</math>.	+	# This means that the function <math>f(x)</math> has a local minimum at <math>x=a</math>.
-	<center>{{:1.3 - Figur - Tangenten till funktion med positiv andraderivata}}</center>	+	<center>{{:1.3 - Figure - The tangent of a function with positive second order derivative}}</center>
	{| width="80%" align="center"		{| width="80%" align="center"
-	||<small>Om derivatan är negativ till vänster om ''x''&nbsp;=&nbsp;''a'' och positiv till höger om ''x''&nbsp;=&nbsp;''a'' så har funktionen ett lokalt minimum i ''x''&nbsp;=&nbsp;''a''.</small>	+	||<small>If the derivative is negative to the left of ''x''&nbsp;=&nbsp;''a'' and positive to the right of ''x''&nbsp;=&nbsp;''a'' the function has a local minimum at ''x''&nbsp;=&nbsp;''a''.</small>
	|}		|}
-	Om <math>f^{\,\prime\prime}(a)=0</math>, får vi ingen information utan ytterligare undersökning krävs, t.ex. teckentabell.	+	If <math>f^{\,\prime\prime}(a)=0</math>, no information can be deduced, and further investigation is required, for example by means of a table of signs. Note in particular that <math>f^{\,\prime\prime}(a)=0</math> does ''not'' imply that the point is a stationary point of inflexion (although <math>f^{\,\prime\prime}(a)=0</math> at all points of inflexion, it can also be zero elsewhere, including at maxima and minima).
	<div class="exempel">		<div class="exempel">
-	'''Exempel 8'''	+	'''Example 8'''
-	Bestäm alla extrempunkter för funktionen <math>f(x)=x^3 -x^2 -x +2</math> och bestäm deras karaktär med hjälp av andraderivatan.	+	Determine all the extrema of the function <math>f(x)=x^3 -x^2 -x +2</math> and determine their character by using the second derivative.
	<br>		<br>
	<br>		<br>
-	Funktionen är ett polynom och är därför deriverbar överallt. Om funktionen har några extrempunkter så måste de därför finnas bland de kritiska punkterna. Vi deriverar därmed funktionen, <math>f^{\,\prime}(x) = 3x^2 -2x - 1</math>, och sätter derivatan lika med noll	+	This function is a polynomial and is therefore differentiable everywhere. If the function has any extrema, they must therefore be found among the stationary points. We thus differentiate the function, <math>f^{\,\prime}(x) = 3x^2 -2x - 1</math>, and equate the derivative to zero
-	{{Fristående formel||<math>	+	{{Displayed math||<math>
	f^{\,\prime}(x) = 0 \quad \Leftrightarrow \quad		f^{\,\prime}(x) = 0 \quad \Leftrightarrow \quad
-	x^2 - \frac{2}{3} x - \frac{1}{3} = 0 \quad \Leftrightarrow \quad	+	x^2 - \tfrac{2}{3} x - \tfrac{1}{3} = 0 \quad \Leftrightarrow \quad
-	x=1 \quad\text{eller}\quad x = -\frac{1}{3}\,\mbox{.}</math>}}	+	x=1 \quad\text{or}\quad x = -\tfrac{1}{3}\,\mbox{.}</math>}}
		+
		+	The function has stationary points at <math>x = 1</math> and <math>x=-\tfrac{1}{3}</math>. By examining the sign of the second derivative <math>f^{\,\prime\prime}(x)=6x-2</math>, we can classify each stationary point .
-	Funktionen har kritiska punkter i <math>x = 1</math> och <math>x=-\tfrac{1}{3}</math>. Med hjälp av tecknet på andraderivatan <math>f^{\,\prime\prime}(x)=6x-2</math> kan vi bestämma vilken typ av extrempunkt respektive kritisk punkt är.	+	* For <math>x=-\tfrac{1}{3}</math> we have that <math>f^{\,\prime\prime}(-\tfrac{1}{3})=-4<0</math> and that means that <math>x=-\tfrac{1}{3}</math> is a local maximum.
-	*För <math>x=-\tfrac{1}{3}</math> har vi att <math>f^{\,\prime\prime}(-\tfrac{1}{3})=-4<0</math> och det betyder att <math>x=-\tfrac{1}{3}</math> är en lokal maximipunkt.	+	* For <math>x=1</math> we have that <math>f^{\,\prime\prime}(1)=4>0</math> and that means that <math>x=1</math> is a local maximum.
-	*För <math>x=1</math> har vi att <math>f^{\,\prime\prime}(1)=4>0</math> och det betyder att <math>x=1</math> är en lokal minimipunkt.	+
	</div>		</div>

---

Current revision

Theory

Exercises

Increasing and decreasing

Informally, a function is increasing if its graph slopes upwards and decreasing if its graph slopes downwards.

The formal mathematical definitions are as follows:

A function is increasing in an interval if for all \displaystyle x_1 and \displaystyle x_2 within the interval

\displaystyle x_1 < x_2\quad\Rightarrow\quad f(x_1) \le f(x_2)\,\mbox{.}

A function is decreasing in an interval if for all \displaystyle x_1 and \displaystyle x_2 within the interval

\displaystyle x_1 < x_2\quad\Rightarrow\quad f(x_1) \ge f(x_2)\,\mbox{.}

In everyday language the definition says, for example, that for an increasing function for any x-value to the right on the x-axis, the value of the function is at least as large as it is for any x-value to the left. Please note that this definition means that a function can be constant in a interval and still be considered to be increasing or decreasing. A function that is constant throughout an interval, according to the definition, is both increasing and decreasing.

If one wants to exclude the possibility of a function being constant on an interval, one talks of strictly increasing and strictly decreasing functions:

A function is strictly increasing in an interval if for all \displaystyle x_1 and \displaystyle x_2 within the interval

\displaystyle x_1 < x_2\quad\Rightarrow\quad f(x_1) < f(x_2)\,\mbox{.}

A function is strictly decreasing in an interval if for \displaystyle x_1 och \displaystyle x_2 within the interval

\displaystyle x_1 < x_2\quad\Rightarrow\quad f(x_1) > f(x_2)\,\mbox{.}

(A strictly increasing or decreasing function cannot be constant in any part of the interval.)

The derivative may of course be used to examine whether a function is increasing or decreasing. We have that

\displaystyle \begin{align*} f^{\,\prime}(x) > 0 \quad&\Rightarrow \quad f(x) \text{ is (strictly) increasing,}\\ f^{\,\prime}(x) < 0 \quad&\Rightarrow \quad f(x) \text{ is (strictly) decreasing.} \end{align*}

Note, however, that this works only one way round. It is entirely possible for a function to be strictly increasing or decreasing on some interval, and for there to be a point, or perhaps more than one, within that interval at which the derivative is zero. As long as any points where the derivative is zero are isolated (that is, provided the derivative isn't zero anywhere close to such points, but only at them), then the function can be strictly increasing or decreasing; problems only arise if the gradient is zero, and the function therefore constant, over some interval.

Stationary points

Points where \displaystyle f^{\,\prime}(x) = 0 are known as stationary points (the term "critical points" is also sometimes used). They are usually one of three kinds:

• Local maximum with \displaystyle f^{\,\prime}(x) > 0 to the left, and \displaystyle f^{\,\prime}(x) < 0 to the right of the point.
• Local minimum with \displaystyle f^{\,\prime}(x) < 0 to the left, and \displaystyle f^{\,\prime}(x) > 0 to the right of the point.
• Stationary point of inflexion with \displaystyle f^{\,\prime}(x) < 0 or \displaystyle f^{\,\prime}(x) > 0 on both sides of the point.

Note that a point may be a local maximum or minimum without \displaystyle f^{\,\prime}(x) = 0; learn more about this in the section on maxima and minima.

Points of inflexion

A point of inflexion is a point where the direction of curvature of a graph changes: where, having curved upwards, it begins curving downwards, or vice versa. In other words, it is a local maximum or minimum for the gradient.

One way to think about points of inflexion is to imagine driving a car along a road shaped like the curve; a point of inflexion is any point at which your steering wheel is exactly centred and you are, for that instant, steering neither right nor left.

At a point of inflexion, the curve's direction of curvature changes; the curve on the left has a stationary point of inflexion in x = 0, where the gradient is zero, but as the other two curves show, not all points of inflexion are stationary points.

We will not study points of inflexion in depth in this section, except to note that a point of inflexion need not necessarily also be a stationary point (though the two can coincide, in which case we would often call the point concerned a "stationary point of inflexion").

The function in the above figure has a local minimum at \displaystyle x = -2, a stationary point of inflexion at \displaystyle x = 0 and a local maximum for \displaystyle x = 2.

Table of signs

By studying the derivative sign (+, - or 0), we can therefore obtain a good idea of the curve's appearance.

One creates a so called table of signs. One first determines the x-values where \displaystyle f^{\,\prime}(x) =0 (together with any where the gradient is undefined) and then calculates the sign of the derivative on either side of these points. With the help of some other "backup" points on the curve and using the table of signs one usually can obtain a satisfactory sketch of the curve.

Maxima and minima (extrema)

A point at which a function takes on its largest or smallest value in comparison with its immediate surroundings is called a local maximum or local minimum (often abbreviated to max and min). Local maxima and minima are together known as extrema.

An extremum may occur in one of three ways:

• At a stationary point (where \displaystyle f^{\,\prime}(x)=0\,).
• At a point where the derivative does not exist (known as a singular point).
• At an endpoint to the interval where the function is defined.

When one is looking for the extrema of a function one must discover and examine all possible candidates for these points. An appropriate working procedures is:

1. Differentiate the function.
2. Check to see if there are any points where \displaystyle f^{\,\prime}(x) is not defined.
3. Determine all points where \displaystyle f^{\,\prime}(x) = 0.
4. Make a table of signs to locate and classify all of the extrema.
5. Calculate the value of the function for all the extrema and at any endpoints.

Global min / max

A function has aglobal maximum at a point if its value there is greater than, or at least equal to, its value at any other point where it is defined; similarly, a global minimum is a point where the function's value is less than, or at most equal to, its value anywhere else.

To determine a function's global max or min one must therefore find all the extrema and calculate the values of the function at them. If the function is defined on an interval with endpoints, one must of course also examine its value at these points.

Note that a function need not have a global max or a global min, even if it has several local extrema.

In applications, circumstances often dictate that a function has a limited interval where it is defined, i.e. one only studies part of the graph of the function. One must therefore be careful in case the global max or min is at an endpoint of the interval.

The above function is only of interest in the interval \displaystyle a\le x \le e. We see that the minimum value of the function in this interval occurs at the stationary point \displaystyle x=b, while the maximim value is found at the endpoint \displaystyle x=e.

The second derivative

The sign of the derivative of a function gives us information about whether the function is increasing or decreasing. Similarly, the sign of the second derivative can show if the first order derivative is increasing or decreasing. This can , among other things, be used to find out whether a given extremum is a maximum or minimum.

If the function \displaystyle f(x) has a stationary point at \displaystyle x=a where \displaystyle f^{\,\prime\prime}(a)<0, then

1. The derivative \displaystyle f^{\,\prime}(x) is strictly decreasing in some interval surrounding \displaystyle x=a.
2. Since \displaystyle f^{\,\prime}(a)=0 then \displaystyle f^{\,\prime}(x)>0 to the left of \displaystyle x=a and \displaystyle f^{\,\prime}(x)<0 to the right of \displaystyle x=a.
3. This means that the function \displaystyle f(x) has a local maximum at \displaystyle x=a.

If the derivative is positive to the left of x = a and negative to the right of x = a the function has a local maximum at x = a.

If the function \displaystyle f(x) has a stationary point at \displaystyle x=a where \displaystyle f^{\,\prime\prime}(a)>0, then

1. The derivative \displaystyle f^{\,\prime}(x) is strictly increasing in some interval around \displaystyle x=a.
2. Since \displaystyle f^{\,\prime}(a)=0 then \displaystyle f^{\,\prime}(x)<0 to the left of \displaystyle x=a and \displaystyle f^{\,\prime}(x)>0 to the right of \displaystyle x=a.
3. This means that the function \displaystyle f(x) has a local minimum at \displaystyle x=a.

If the derivative is negative to the left of x = a and positive to the right of x = a the function has a local minimum at x = a.

If \displaystyle f^{\,\prime\prime}(a)=0, no information can be deduced, and further investigation is required, for example by means of a table of signs. Note in particular that \displaystyle f^{\,\prime\prime}(a)=0 does not imply that the point is a stationary point of inflexion (although \displaystyle f^{\,\prime\prime}(a)=0 at all points of inflexion, it can also be zero elsewhere, including at maxima and minima).