1.3 Maximierungs- und Minimierungsprobleme
Aus Online Mathematik Brückenkurs 2
K |
(Edited for local terminology and idiom. "Stationary points", "maxima", "minima" and "extrema" used throughout; section on points of inflexion added (this section needs a figure).) |
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After this section, you will have learned: | After this section, you will have learned: | ||
| - | * The definition of strictly increasing | + | * The definition of strictly increasing, strictly decreasing, 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. | * 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 | + | * To locate stationary points and, by studying the sign of the derivative, classify them as local maxima, local minima, and stationary points of inflexion. |
* To sketch the graph of a function by constructing a table of signs for the derivative. | * To sketch the graph of a function by constructing a table of signs for the derivative. | ||
| - | * To determine | + | * To determine global and local maxima and minima by 1) studying the sign of the derivative, 2) considering points where the function is not differentiable, 3) examining the endpoints of the interval where the function is defined. |
| - | * To | + | * To use the sign of the second derivative to distinguish between local maxima and local minima. |
}} | }} | ||
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== Increasing and decreasing == | == Increasing and decreasing == | ||
| - | + | Informally, a function is increasing if its graph slopes upwards and decreasing if its graph slopes downwards. | |
| - | The mathematical definitions are as follows: | + | The formal 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 | A function is increasing in an interval if for all <math>x_1</math> and <math>x_2</math> within the interval | ||
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{{Displayed math||<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>}} | ||
| - | 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 | + | 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 | + | 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 <math>x_1</math> and <math>x_2</math> within the interval | A function is ''strictly'' increasing in an interval if for all <math>x_1</math> and <math>x_2</math> within the interval | ||
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{{Displayed math||<math>x_1 < x_2\quad\Rightarrow\quad f(x_1) > f(x_2)\,\mbox{.}</math>}} | {{Displayed math||<math>x_1 < x_2\quad\Rightarrow\quad f(x_1) > f(x_2)\,\mbox{.}</math>}} | ||
| - | (A strictly increasing | + | (A strictly increasing or decreasing function cannot be constant in any part of the interval.) |
<div class="exempel"> | <div class="exempel"> | ||
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{{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>}} | {{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>}} | ||
| - | Note that | + | 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 <math>f^{\,\prime}(x) = 0</math> are known as | + | 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: |
* 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 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. | * 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. | ||
| - | * | + | * Stationary point of inflexion 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)| | + | 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]]''. |
| + | |||
| + | == 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. | ||
| + | |||
| + | 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> | <center>{{:1.3 - Figure - The graph of f(x) = x³ - x⁵}}</center> | ||
| - | The function in the above figure has a local minimum | + | 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>. |
== Table of signs == | == Table of signs == | ||
| - | By studying the derivative sign (+, - or 0), we | + | 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 <math>f^{\,\prime}(x) =0</math> and then calculates the sign of the derivative on | + | 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"> | ||
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|} | |} | ||
| - | In the table's last line, we have given arrows that indicate whether the function is strictly increasing <math>(\,\nearrow\,\,)</math> or strictly decreasing <math>(\,\searrow\,\,)</math> in each interval as well as the value of the function value at the | + | In the table's last line, we have given arrows that indicate whether the function is strictly increasing <math>(\,\nearrow\,\,)</math> or strictly decreasing <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>. |
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: | 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: | ||
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| - | == | + | == Maxima and minima (extrema) == |
| - | A point | + | 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 | + | :* At a stationary point (where <math>f^{\,\prime}(x)=0\,</math>). |
:* At a point where the derivative does not exist (known as a ''singular point''). | :* 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. | :* At an endpoint to the interval where the function is defined. | ||
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''' Example 3''' | ''' Example 3''' | ||
| - | For the function below there are four | + | 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 - Figure - The graph of f(x), where f has extreme points in 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> | ||
| - | 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> | + | 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. |
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. | 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. | ||
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</div> | </div> | ||
| - | When one is looking for | + | 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: |
:# Differentiate the function. | :# Differentiate the function. | ||
:# Check to see if there are any points where <math>f^{\,\prime}(x)</math> is not defined. | :# 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>. | :# Determine all points where <math>f^{\,\prime}(x) = 0</math>. | ||
| - | :# Make a table of signs to | + | :# Make a table of signs to locate and classify all of the extrema. |
| - | :# Calculate the value of the function for all the | + | :# Calculate the value of the function for all the extrema and at any endpoints. |
<div class="exempel"> | <div class="exempel"> | ||
''' Example 4''' | ''' Example 4''' | ||
| - | Determine all the | + | Determine all the extrema of the curve <math>y=3x^4 +4x^3 - 12x^2 + 12</math>. |
<br> | <br> | ||
<br> | <br> | ||
| - | The | + | The function's derivative is given by |
{{Displayed math||<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>}} | ||
| - | In order to determine how the sign of the derivative varies along the real-number axis, we factorise the derivative as | + | 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 |
{{Displayed math||<math> | {{Displayed math||<math> | ||
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|} | |} | ||
| - | 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. | + | 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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|} | |} | ||
| - | The curve has thus local | + | 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> | ||
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''' Example 5''' | ''' Example 5''' | ||
| - | Determine all | + | Determine all extrema of the curve <math>y= x - x^{2/3}</math>. |
<br> | <br> | ||
<br> | <br> | ||
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{{Displayed math||<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} | ||
| - | \ | + | \, \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 | + | 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>. |
| - | The | + | The stationary points of the function are given by |
{{Displayed math||<math> | {{Displayed math||<math> | ||
y'=0 \quad \Leftrightarrow \quad | y'=0 \quad \Leftrightarrow \quad | ||
| - | 1= \frac {2}{3} \ | + | 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>}} | ||
| - | The only points | + | 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" | ||
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| - | == | + | == Global min / max == |
| - | A function has | + | 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. |
| - | To determine a | + | 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 | + | 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"> | ||
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<center>{{:1.3 - Figure - A function with local and global extreme points}}</center> | <center>{{:1.3 - Figure - A function with local and global extreme points}}</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 | + | 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>. |
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<br> | <br> | ||
<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 | + | 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 |
{{Displayed math||<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>}} | ||
| - | The point <math>x = –1</math> is outside the interval | + | 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, |
{{Displayed math||<math>\begin{align*} f(-0\textrm{.}5) &= 3\textrm{.}375\,\mbox{,}\\[4pt] f(1)&=0\,\mbox{.} \end{align*}</math>}} | {{Displayed math||<math>\begin{align*} f(-0\textrm{.}5) &= 3\textrm{.}375\,\mbox{,}\\[4pt] f(1)&=0\,\mbox{.} \end{align*}</math>}} | ||
| - | The | + | 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 - Figure - The graph of f(x) = x³ - 3x + 2}}</center> | <center>{{:1.3 - Figure - The graph of f(x) = x³ - 3x + 2}}</center> | ||
| - | The figure shows the function with the whole graph as a dashed curve , | + | 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> | ||
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== The second derivative == | == The second derivative == | ||
| - | The sign of the derivative of a function gives us information about whether the function is increasing or decreasing. Similarly, the | + | 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 <math>f(x)</math> has a | + | 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 |
# The derivative <math>f^{\,\prime}(x)</math> is strictly decreasing in some interval surrounding <math>x=a</math>. | # 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>. | # 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>. | ||
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| - | If the function <math>f(x)</math> has a | + | 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 |
# The derivative <math>f^{\,\prime}(x)</math> is strictly increasing in some interval around <math>x=a</math>. | # 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>. | # 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>. | ||
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| - | If <math>f^{\,\prime\prime}(a)=0</math>, no information can be deduced, | + | 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). |
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'''Example 8''' | '''Example 8''' | ||
| - | Determine all the | + | 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> | ||
| - | This function is a polynomial and is therefore differentiable everywhere. If the function has | + | 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 |
{{Displayed math||<math> | {{Displayed math||<math> | ||
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x=1 \quad\text{or}\quad x = -\tfrac{1}{3}\,\mbox{.}</math>}} | x=1 \quad\text{or}\quad x = -\tfrac{1}{3}\,\mbox{.}</math>}} | ||
| - | The function has | + | 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 . |
* 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=-\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. | ||
Version vom 13:38, 7. Jan. 2009
| Theory | Exercises |
Contents:
- Sketching curves
- Maximum and minimum problems
Learning outcomes:
After this section, you will have learned:
- The definition of strictly increasing, strictly decreasing, local maximum, local minimum, global maximum, global minimum.
- That if \displaystyle f^{\,\prime}>0 in an interval then \displaystyle f is strictly increasing in the interval, and that if \displaystyle f^{\,\prime}<0 in an interval then \displaystyle f is strictly decreasing in the interval.
- To locate stationary points and, by studying the sign of the derivative, classify them as local maxima, local minima, and stationary points of inflexion.
- To sketch the graph of a function by constructing a table of signs for the derivative.
- To determine global and local maxima and minima by 1) studying the sign of the derivative, 2) considering points where the function is not differentiable, 3) examining the endpoints of the interval where the function is defined.
- To use the sign of the second derivative to distinguish between local maxima and local minima.
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.)
Example 1
- The function \displaystyle y= f(x) whose graph is given in the chart below on the far left is increasing in the interval \displaystyle 0 \le x \le 6.
- The function \displaystyle y=-x^3\!/4 is a strictly decreasing function.
- The function \displaystyle y=x^2 is a strictly increasing function for \displaystyle x \ge 0.
| 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) = x² | ||
| Graph of the function in part a | Graph of the function f(x) = - x³/4 | Graph of the function f(x) = x² |
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.
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.
Example 2
Make a table of signs of the derivative of the function \displaystyle f(x) = x^3 -12x + 6 and then sketch the graph of the function.
The functions derivative is given by
| \displaystyle
f^{\,\prime}(x) = 3x^2 -12 = 3(x^2-4) = 3(x-2)(x+2). |
The factor \displaystyle x-2 is negative to the left of \displaystyle x=2 and positive to the right of \displaystyle x=2. In the same way the factor \displaystyle x+2 is negative to the left of \displaystyle x=-2 and positive to the right of \displaystyle x=-2. This information can be summarised in a table:
| \displaystyle x | \displaystyle -2 | \displaystyle 2 | |||
| \displaystyle x-2 | \displaystyle - | \displaystyle - | \displaystyle - | \displaystyle 0 | \displaystyle + |
| \displaystyle x+2 | \displaystyle - | \displaystyle 0 | \displaystyle + | \displaystyle + | \displaystyle + |
Since the derivative is the product of \displaystyle x-2 and \displaystyle x+2 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 :
| \displaystyle x | \displaystyle -2 | \displaystyle 2 | |||
| \displaystyle f^{\,\prime}(x) | \displaystyle + | \displaystyle 0 | \displaystyle - | \displaystyle 0 | \displaystyle + |
| \displaystyle f(x) | \displaystyle \nearrow | \displaystyle 22 | \displaystyle \searrow | \displaystyle -10 | \displaystyle \nearrow |
In the table's last line, we have given arrows that indicate whether the function is strictly increasing \displaystyle (\,\nearrow\,\,) or strictly decreasing \displaystyle (\,\searrow\,\,) in each interval as well as the value of the function value at the stationary points \displaystyle x=-2 and \displaystyle x=2.
From the figure, we see that \displaystyle f(x) has a local maximum at \displaystyle (–2, 22) and a local minimum at \displaystyle (2, –10). The graph now can be sketched:
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.
Example 3
For the function below there are four extrema: maximum at \displaystyle x=c and \displaystyle x=e, and minimum at \displaystyle x=a and \displaystyle x=d.
At \displaystyle x=a, \displaystyle x=b and \displaystyle x=d one has \displaystyle f^{\,\prime}(x) =0, but it is only at \displaystyle x=a and \displaystyle x=d that there are extrema, since \displaystyle x=b is a stationary point of inflexion.
At \displaystyle x=c 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 \displaystyle x=e is an endpoint.
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:
- Differentiate the function.
- Check to see if there are any points where \displaystyle f^{\,\prime}(x) is not defined.
- Determine all points where \displaystyle f^{\,\prime}(x) = 0.
- Make a table of signs to locate and classify all of the extrema.
- Calculate the value of the function for all the extrema and at any endpoints.
Example 4
Determine all the extrema of the curve \displaystyle y=3x^4 +4x^3 - 12x^2 + 12.
The function's derivative is given by
| \displaystyle
y' = 12x^3 + 12x^2 - 24x = 12x(x^2+x-2)\,\mbox{.} |
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 \displaystyle 12x and we can factorise further the remaining term \displaystyle x^2+x-2 by identifying its zeros
| \displaystyle
x^2+x-2=0\qquad\Leftrightarrow\qquad x=-2\quad\text{or}\quad x=1. |
This means that \displaystyle x^2+x-2=(x+2)(x-1) and the derivative can be rewritten as
| \displaystyle y' = 12x(x+2)(x-1)\,\mbox{.} |
It can be seen immediately from this that the derivative is zero for \displaystyle x=-2, \displaystyle x=0 and \displaystyle x=1. 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 \displaystyle x.
| \displaystyle x | \displaystyle -2 | \displaystyle 0 | \displaystyle 1 | ||||
| \displaystyle x+2 | \displaystyle - | \displaystyle 0 | \displaystyle + | \displaystyle + | \displaystyle + | \displaystyle + | \displaystyle + |
| \displaystyle x | \displaystyle - | \displaystyle - | \displaystyle - | \displaystyle 0 | \displaystyle + | \displaystyle + | \displaystyle + |
| \displaystyle x-1 | \displaystyle - | \displaystyle - | \displaystyle - | \displaystyle - | \displaystyle - | \displaystyle 0 | \displaystyle + |
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.
| \displaystyle x | \displaystyle -2 | \displaystyle 0 | \displaystyle 1 | ||||
| \displaystyle f^{\,\prime}(x) | \displaystyle - | \displaystyle 0 | \displaystyle + | \displaystyle 0 | \displaystyle - | \displaystyle 0 | \displaystyle + |
| \displaystyle f(x) | \displaystyle \searrow | \displaystyle -20 | \displaystyle \nearrow | \displaystyle 12 | \displaystyle \searrow | \displaystyle 7 | \displaystyle \nearrow |
The curve has thus local minima at \displaystyle (–2, –20) and \displaystyle (1, 7) and a local maximum at \displaystyle (0, 12).
Example 5
Determine all extrema of the curve \displaystyle y= x - x^{2/3}.
The derivative of the function is given by
| \displaystyle
y' = 1 - \frac{2}{3} x^{-1/3} = 1- \frac {2}{3} \, \frac{1}{\sqrt[\scriptstyle 3]{x}}\,\mbox{.} |
From this expression, we see that \displaystyle y' is not defined for \displaystyle x = 0 (although which \displaystyle y is defined). This means that the function has a singular point at \displaystyle x=0.
The stationary points of the function are given by
| \displaystyle
y'=0 \quad \Leftrightarrow \quad 1= \frac {2}{3} \, \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{.} |
The only points at which the function might have an extremum are thus \displaystyle x=0 and \displaystyle x=\tfrac{8}{27}. In order to determine the nature of these points we create a table of signs:
| \displaystyle x | \displaystyle 0 | \displaystyle \frac{8}{27} | |||
| \displaystyle y' | \displaystyle + | not def. | \displaystyle - | \displaystyle 0 | \displaystyle + |
| \displaystyle y | \displaystyle \nearrow | \displaystyle 0 | \displaystyle \searrow | \displaystyle -\frac{4}{27} | \displaystyle \nearrow |
The curve has a local maximum at \displaystyle (0, 0) (a cusp) and a local minimum at \displaystyle (\tfrac{8}{27},-\tfrac{4}{27})\,.
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.
Example 6
In the first figure the function has no global maximum nor global minimum. In the second figure the function has no global minimum.
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.
Example 7
Determine the maximum and minimum value of the function \displaystyle f(x) = x^3 -3x + 2 in the interval \displaystyle -0\textrm{.}5 \le x \le 1\,.
We differentiate the function, \displaystyle f^{\,\prime}(x) = 3x^2 -3, and put the derivative equal to zero to obtain all the stationary points
| \displaystyle f^{\,\prime}(x) = 0 \quad \Leftrightarrow \quad x^2 = 1 \quad \Leftrightarrow \quad x= \pm 1\,\mbox{.} |
The point \displaystyle x = –1 is outside the interval on which the function is defined, and \displaystyle x = 1 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,
| \displaystyle \begin{align*} f(-0\textrm{.}5) &= 3\textrm{.}375\,\mbox{,}\\[4pt] f(1)&=0\,\mbox{.} \end{align*} |
The function's maximum value on the given interval is thus \displaystyle 3\textrm{.}375. The minimum value is \displaystyle 0 (see the figure).
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.
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
- The derivative \displaystyle f^{\,\prime}(x) is strictly decreasing in some interval surrounding \displaystyle x=a.
- 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.
- 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
- The derivative \displaystyle f^{\,\prime}(x) is strictly increasing in some interval around \displaystyle x=a.
- 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.
- 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).
Example 8
Determine all the extrema of the function \displaystyle f(x)=x^3 -x^2 -x +2 and determine their character by using the second derivative.
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, \displaystyle f^{\,\prime}(x) = 3x^2 -2x - 1, and equate the derivative to zero
| \displaystyle
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{.} |
The function has stationary points at \displaystyle x = 1 and \displaystyle x=-\tfrac{1}{3}. By examining the sign of the second derivative \displaystyle f^{\,\prime\prime}(x)=6x-2, we can classify each stationary point .
- For \displaystyle x=-\tfrac{1}{3} we have that \displaystyle f^{\,\prime\prime}(-\tfrac{1}{3})=-4<0 and that means that \displaystyle x=-\tfrac{1}{3} is a local maximum.
- For \displaystyle x=1 we have that \displaystyle f^{\,\prime\prime}(1)=4>0 and that means that \displaystyle x=1 is a local maximum.
