From Förberedande kurs i matematik 2

The action you have requested is limited to users in one of the groups Sysops, approved.

You can view and copy the source of this page:

__NOTOC__ {| border="0" cellspacing="0" cellpadding="0" height="30" width="100%" | style="border-bottom:1px solid #797979" width="5px" | &nbsp; {{Vald flik|[[1.3 Max- och minproblem|Theory]]}} {{Ej vald flik|[[1.3 Övningar|Exercises]]}} | style="border-bottom:1px solid #797979" width="100%"| &nbsp; |} {{Info| '''Contents:''' * Sketching curves * Maximum and minimum problems }} {{Info| '''Learning outcomes:''' After this section, you will have learned: * The definition of strictly increasing functions, strictly decreasing functions, local maximum, local minimum, global maximum, global minimum. * 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. * To determine local maximums and minimums and points of inflection by studying the sign of the derivative. * To sketch the graph of a function by constructing a table of signs for the derivative. * To determine the global and local maximums and minimums by 1)&nbsp;studying the sign of the derivative, 2)&nbsp;points where the function is not differentiable, 3)&nbsp;the endpoints of the interval where the function is defined. * To distinguish between local maximums and minimums from the the sign of the second order derivative. }} == Increasing and decreasing == The concepts of increasing and decreasing may seem obvious when talking about mathematical functions, if the function is increasing then the graph slopes upwards and if it is decreasing it slopes downwards. The mathematical definitions are as follows: 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>}} 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>}} 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 that an increasing / decreasing function is constant in an interval one rather talks of ''strictly'' increasing and ''strictly'' decreasing functions: A function is ''strictly'' 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) < 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 {{Fristående formel||<math>x_1 < x_2\quad\Rightarrow\quad f(x_1) > f(x_2)\,\mbox{.}</math>}} (A strictly increasing / decreasing function cannot be constant in any part of the interval.) <div class="exempel"> '''Example 1''' <ol type="a"> <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> <li> The function <math>y=-x^3\!/4</math> is a strictly decreasing function.</li> <li> The function <math>y=x^2</math> is a strictly increasing function for <math>x \ge 0</math>.</li> </ol> {| align="center" |- ||{{:1.3 - Figur - Grafen till f(x), där f är styckvis linjär och konstant}} | width="10px" | ||{{:1.3 - Figur - Grafen f(x) = -x³/4}} | width="10px"| ||{{:1.3 - Figur - Grafen till f(x) = x²}} |- ||<small>Graph of the function in part <br> &nbsp;a</small> || ||<small>Graph of the function<br> ''f''(''x'')&nbsp;=&nbsp;-&nbsp;''x''³/4</small> || ||<small>Graph of the function<br> ''f''(''x'')&nbsp;=&nbsp;x²</small> |} </div> 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{ is (strictly) increasing,}\\ f^{\,\prime}(x) < 0 \quad&\Rightarrow \quad f(x) \text{ is (strictly) decreasing.} \end{align*}</math>}} Note that even isolated '''points''' where <math>f^{\,\prime}(x) = 0</math> can be part of a strictly increasing or decreasing interval. == Critical points == Points where <math>f^{\,\prime}(x) = 0</math> are known as critical (or stationary) points and are usually one of three kinds: * 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. * 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. * Point of inflection with <math>f^{\,\prime}(x) < 0</math> or <math>f^{\,\prime}(x) > 0</math> on both sides of the point. 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)|maximums and minimums]]''. <center>{{:1.3 - Figur - Grafen till f(x) = x³ - x⁵}}</center> The function in the above figure has a local minimum for <math>x = -2</math>, a point of inflection for <math>x = 0</math> and a local maximum for <math>x = 2</math>. == Table of signs == By studying the derivative sign (+, - or 0), we therefore can obtain a good idea of the curve's appearance. One creates a so called ''table of signs''. One first determines the ''x''-values where <math>f^{\,\prime}(x) =0</math> and then calculates the sign of the derivative on both sides 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"> '''Example 2''' 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> The functions derivative is given by {{Fristående formel||<math> f^{\,\prime}(x) = 3x^2 -12 = 3(x^2-4) = 3(x-2)(x+2).</math>}} 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: {| border="1" cellpadding="5" cellspacing="0" align="center" |- |width="50px" align="center" style="background:#efefef;"| <math>x</math> |style="background:#efefef;"| |width="50px" align="center" style="background:#efefef;"| <math>-2</math> |style="background:#efefef;"| |width="50px" align="center" style="background:#efefef;"| <math>2</math> |style="background:#efefef;"| |- |width="50px" align="center"| <math>x-2</math> |width="50px" align="center"| <math>-</math> |width="50px" align="center"| <math>-</math> |width="50px" align="center"| <math>-</math> |width="50px" align="center"| <math>0</math> |width="50px" align="center"| <math>+</math> |- |width="50px" align="center"| <math>x+2</math> |width="50px" align="center"| <math>-</math> |width="50px" align="center"| <math>0</math> |width="50px" align="center"| <math>+</math> |width="50px" align="center"| <math>+</math> |width="50px" align="center"| <math>+</math> |- |} 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 : {| border="1" cellpadding="5" cellspacing="0" align="center" |- |width="50px" align="center" style="background:#efefef;"| <math>x</math> |width="50px" align="center" style="background:#efefef;"| |width="50px" align="center" style="background:#efefef;"| <math>-2</math> |width="50px" align="center" style="background:#efefef;"| |width="50px" align="center" style="background:#efefef;"| <math>2</math> |width="50px" align="center" style="background:#efefef;"| |- |width="50px" align="center"| <math>f^{\,\prime}(x)</math> |width="50px" align="center"| <math>+</math> |width="50px" align="center"| <math>0</math> |width="50px" align="center"| <math>-</math> |width="50px" align="center"| <math>0</math> |width="50px" align="center"| <math>+</math> |- |width="50px" align="center"| <math>f(x)</math> |width="50px" align="center"| <math>\nearrow</math> |width="50px" align="center"| <math>22</math> |width="50px" align="center"| <math>\searrow</math> |width="50px" align="center"| <math>-10</math> |width="50px" align="center"| <math>\nearrow</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 critical points <math>x=-2</math> and <math>x=2</math>. 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> </div> == Maximums and minimums (extreme points) == A point in which a function takes on its largest or smallest value in comparison with its surroundings is called a ''local maximum '' or ''local minimum'' (often abbreviated to max and min). A joint name is ''extreme point'' or just ''extreme''. A extreme may occur in one of three ways: :* At a critical point (where <math>f^{\,\prime}(x)=0\,</math>). :* 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. <div class="exempel"> ''' Example 3''' For the function below there are four extreme points: 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> 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> one has extreme points, since <math>x=b</math> is a point of inflection. 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> When one is looking for extreme points of a function one must discover and examine all possible candidates for these points. An appropriate working procedures is: :# Differentiate the function. :# Check to see if there are any points where <math>f^{\,\prime}(x)</math> is not defined. :# Determine all points where <math>f^{\,\prime}(x) = 0</math>. :# Make a table of signs to get all of the extreme points. :# Calculate the value of the function for all the extreme points and at any endpoints. <div class="exempel"> ''' Example 4''' Determine all the extreme points on the curve <math>y=3x^4 +4x^3 - 12x^2 + 12</math>. <br> <br> The functions derivative is given by {{Fristående formel||<math> y' = 12x^3 + 12x^2 - 24x = 12x(x^2+x-2)\,\mbox{.}</math>}} In order to determine how the sign of the derivative varies along the real-number axis, we factorise the derivative as far as possible. We have already managed to factorise out <math>12x</math> and we can factorise further the remaining term <math>x^2+x-2</math> by identifying its zeros {{Fristående formel||<math> x^2+x-2=0\qquad\Leftrightarrow\qquad x=-2\quad\text{or}\quad x=1.</math>}} 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>}} 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>. {| border="1" cellpadding="5" cellspacing="0" align="center" |- |width="40px" align="center" style="background:#efefef;"| <math>x</math> |width="40px" align="center" style="background:#efefef;"| |width="40px" align="center" style="background:#efefef;"| <math>-2</math> |width="40px" align="center" style="background:#efefef;"| |width="40px" align="center" style="background:#efefef;"| <math>0</math> |width="40px" align="center" style="background:#efefef;"| |width="40px" align="center" style="background:#efefef;"| <math>1</math> |width="40px" align="center" style="background:#efefef;"| |- |width="40px" align="center"| <math>x+2</math> |width="40px" align="center"| <math>-</math> |width="40px" align="center"| <math>0</math> |width="40px" align="center"| <math>+</math> |width="40px" align="center"| <math>+</math> |width="40px" align="center"| <math>+</math> |width="40px" align="center"| <math>+</math> |width="40px" align="center"| <math>+</math> |- |width="40px" align="center"| <math>x</math> |width="40px" align="center"| <math>-</math> |width="40px" align="center"| <math>-</math> |width="40px" align="center"| <math>-</math> |width="40px" align="center"| <math>0</math> |width="40px" align="center"| <math>+</math> |width="40px" align="center"| <math>+</math> |width="40px" align="center"| <math>+</math> |- |width="40px" align="center"| <math>x-1</math> |width="40px" align="center"| <math>-</math> |width="40px" align="center"| <math>-</math> |width="40px" align="center"| <math>-</math> |width="40px" align="center"| <math>-</math> |width="40px" align="center"| <math>-</math> |width="40px" align="center"| <math>0</math> |width="40px" align="center"| <math>+</math> |- |} 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" |- |width="40px" align="center" style="background:#efefef;"| <math>x</math> |width="40px" align="center" style="background:#efefef;"| |width="40px" align="center" style="background:#efefef;"| <math>-2</math> |width="40px" align="center" style="background:#efefef;"| |width="40px" align="center" style="background:#efefef;"| <math>0</math> |width="40px" align="center" style="background:#efefef;"| |width="40px" align="center" style="background:#efefef;"| <math>1</math> |width="40px" align="center" style="background:#efefef;"| |- |width="40px" align="center"| <math>f^{\,\prime}(x)</math> |width="40px" align="center"| <math>-</math> |width="40px" align="center"| <math>0</math> |width="40px" align="center"| <math>+</math> |width="40px" align="center"| <math>0</math> |width="40px" align="center"| <math>-</math> |width="40px" align="center"| <math>0</math> |width="40px" align="center"| <math>+</math> |- |width="40px" align="center"| <math>f(x)</math> |width="40px" align="center"| <math>\searrow</math> |width="40px" align="center"| <math>-20</math> |width="40px" align="center"| <math>\nearrow</math> |width="40px" align="center"| <math>12</math> |width="40px" align="center"| <math>\searrow</math> |width="40px" align="center"| <math>7</math> |width="40px" align="center"| <math>\nearrow</math> |- |} The curve has thus local minimums at <math>(–2, –20)</math> and <math>(1, 7)</math> and a local maximum at <math>(0, 12)</math>. </div> <div class="exempel"> ''' Example 5''' Determine all extreme points for the curve <math>y= x - x^{2/3}</math>. <br> <br> The derivative of the function is given by {{Fristående formel||<math> y' = 1 - \frac{2}{3} x^{-1/3} = 1- \frac {2}{3} \cdot \frac{1}{\sqrt[\scriptstyle 3]{x}}\,\mbox{.}</math>}} From this expression, we see that <math>y'</math> is not defined for <math>x = 0</math> (which <math>y</math> is however). This means that the function has a singular point at <math>x=0</math>. The critical points for the function are given by {{Fristående formel||<math> y'=0 \quad \Leftrightarrow \quad 1= \frac {2}{3} \cdot \frac{1}{\sqrt[3]{x}}\quad\Leftrightarrow\quad \sqrt[3]{x} = \tfrac {2}{3}\quad \Leftrightarrow \quad x = \bigl(\tfrac{2}{3}\bigr)^3 = \tfrac{8}{27}\,\mbox{.}</math>}} The only points for which the function might have extreme points 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" |- |width="50px" align="center" style="background:#efefef;"| <math>x</math> |width="50px" align="center" style="background:#efefef;"| |width="50px" align="center" style="background:#efefef;"| <math>0</math> |width="50px" align="center" style="background:#efefef;"| |width="50px" align="center" style="background:#efefef;"| <math>\frac{8}{27}</math> |width="50px" align="center" style="background:#efefef;"| |- |width="50px" align="center"| <math>y'</math> |width="50px" align="center"| <math>+</math> |width="50px" align="center"| not def. |width="50px" align="center"| <math>-</math> |width="50px" align="center"| <math>0</math> |width="50px" align="center"| <math>+</math> |- |width="50px" align="center"| <math>y</math> |width="50px" align="center"| <math>\nearrow</math> |width="50px" align="center"| <math>0</math> |width="50px" align="center"| <math>\searrow</math> |width="50px" align="center"| <math>-\frac{4}{27}</math> |width="50px" align="center"| <math>\nearrow</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> </div> == Absolute min / max == A function has an ''absolute'' (or ''global'') maximum (minimum) at a point if the value of the function is not greater (less) in any other point in the interval where the function is defined. This often is called the functions largest (least) value. To determine a functions absolute max or min one must therefore find all the extreme points and calculate the values of the function at them. If the interval where the function is defined has endpoints, one must of course examine the function value at these points. Note that an function may lack both an absolute max and an absolute min. Note also that a function can have several local extreme points without having a global max or min. <div class="exempel"> ''' Example 6''' <center>{{:1.3 - Figur - Två funktioner som saknar min och max resp. min}}</center> In the first figure the function has no global maximum nor global minimum. In the second figure the function has no global minimum. </div> 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> 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 critical point, <math>x=b</math>, while the maximim value is found at the endpoint <math>x=e</math>. <div class="exempel"> ''' Example 7''' 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> We differentiate the function, <math>f^{\,\prime}(x) = 3x^2 -3</math>, and put the derivative equal to zero to obtain all the critical points {{Fristående formel||<math>f^{\,\prime}(x) = 0 \quad \Leftrightarrow \quad x^2 = 1 \quad \Leftrightarrow \quad x= \pm 1\,\mbox{.}</math>}} The point <math>x = –1</math> is outside the interval where the function is defined and <math>x = 1</math> lies on the interval where the function is defined at one endpoint. Since the function has no singular points (the function is differentiable everywhere) the functions maximum and minimum must be at the intervals endpoints, {{Fristående formel||<math>\begin{align*} f(-0\textrm{.}5) &= 3\textrm{.}375\,\mbox{,}\\[4pt] f(1)&=0\,\mbox{.} \end{align*}</math>}} The functions maximum value in 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> The figure shows the function with the whole graph as a dashed curve , and with the part that is within the given interval as a continuous curve. </div> == The second derivative == The sign of the derivative of a function gives us information about whether the function is increasing or decreasing. Similarly, the second order derivatives sign can show if the first order derivative is increasing or decreasing. This can , among other things, be used to find out whether a given extreme point is a maximum or minimum. If the function <math>f(x)</math> has a critical point at <math>x=a</math> where <math>f^{\,\prime\prime}(a)<0</math>, then # The derivative <math>f^{\,\prime}(x)</math> is strictly decreasing in some interval surrounding <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>. # 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> {| width="80%" align="center" ||<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> |} If the function <math>f(x)</math> has a critical point at <math>x=a</math> where <math>f^{\,\prime\prime}(a)>0</math>, then # The derivative <math>f^{\,\prime}(x)</math> is strictly increasing in some interval around <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>. # 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> {| width="80%" align="center" ||<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> |} If <math>f^{\,\prime\prime}(a)=0</math>, no information can be deduced, but further investigation is required, for example by means of a table of signs. <div class="exempel"> '''Example 8''' Determine all the extreme points for the function <math>f(x)=x^3 -x^2 -x +2</math> and determine their character by using the second order derivative. <br> <br> This function is a polynomial and is therefore differentiable everywhere. If the function has some extreme points they must therefore be found among the critical points. We thus differentiate the function, <math>f^{\,\prime}(x) = 3x^2 -2x - 1</math>, and equate the derivative to zero {{Fristående formel||<math> f^{\,\prime}(x) = 0 \quad \Leftrightarrow \quad x^2 - \tfrac{2}{3} x - \tfrac{1}{3} = 0 \quad \Leftrightarrow \quad x=1 \quad\text{or}\quad x = -\tfrac{1}{3}\,\mbox{.}</math>}} The function has critical points at <math>x = 1</math> and <math>x=-\tfrac{1}{3}</math>. By examining the sign of the second order derivative <math>f^{\,\prime\prime}(x)=6x-2</math> we can classify the extreme points for each critical point . * 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. * 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. </div>

• 1.3 - Figure - A function with local and global extreme points
• 1.3 - Figure - Different types of inflexion points
• 1.3 - Figure - The graph of f(x), where f has extreme points in x = a, b, c, d, e
• 1.3 - Figure - The graph of f(x), where f is piecewise linear and constant
• 1.3 - Figure - The graph of f(x) = -x³/4
• 1.3 - Figure - The graph of f(x) = x - x^⅔
• 1.3 - Figure - The graph of f(x) = x²
• 1.3 - Figure - The graph of f(x) = x³ - 12x + 6
• 1.3 - Figure - The graph of f(x) = x³ - 3x + 2
• 1.3 - Figure - The graph of f(x) = x³ - x⁵
• 1.3 - Figure - The tangent of a function with negative second order derivative
• 1.3 - Figure - The tangent of a function with positive second order derivative
• 1.3 - Figure - Two functions where one has no min nor max, and one has no min
• Template:Displayed math
• Template:Info
• Template:Not selected tab
• Template:Selected tab

Return to Main Page.