Local minima and maxima

Applications of differentiation: Analysis of functions

Local minima and maxima

Let's have another look at an exercise from week 4.

Calculate the tangent $l(x)$ of $h(x)=-x^2+8\cdot x+8$ and at the point $\rv{4,24}$ .

$l(x)= 24$ .

The function rule of the tangent line has the form $l(x)=a\cdot x+b$ . S o need to find the values for $a$ and $b$ .

The slope $a$ of the tangent has to be $a=f'(4)$ . The derivative of $f$ is $f'(x)=8-2\cdot x$ . This means that $a=f'(4)=0$ .

The formula of the tangent therefore looks like this: $l(x)=0 \cdot x +b$ . You now only have to calculate $b$ . This is simple. The tangent goes through $\rv{4,24}$ . Hence, $b=24-0 \cdot 4 = 24$ . We conclude that the function rule for the tangent is $l(x)=24$ .

New example

You can see that the tangent line is a horizontal line. If a function changes from increasing to decreasing (or from decreasing to increasing), then its derivative changes signs, so the derivative at the transition point will be $0$ .

We have seen that as a function decreases, its derivative is negative and that as a function increases its derivative is positive. We can also reverse this.

Stationary points

Let $f$ be a differentiable function. A point $x$ with $f'(x)=0$ is called a stationary point of said $f$ .

Stationary points say something about how a function behaves, because a function can only change from increasing to decreasing, and vice versa, in a stationary point.

First, we will now take a look at the definition of a local maximum and minimum before we look at how we can use the stationary point to find these extremes.

We can say that $f$ has a local maximum in $p$ if there is an open interval $\ivoo{c}{d}$ around $p$ with $f(x)\le f(p)$ for all $x\in\ivoo{c}{d}\cap I$ .

We can say that $f$ has a local minimum in $p$ if there is an open interval $\ivoo{c}{d}$ around $p$ with $f(x)\ge f(p)$ for all $x\in\ivoo{c}{d}\cap I$ .

All local minima and maxima together we call extreme points.

Points at the edge of $I$ can also be local maximums. If a local maximum is not on the edge of $I$ , then the open interval $\ivoo{c}{d}$ can be chosen completely in $I$ .

Local extrema, stationary points

If $I$ is an open interval, $f$ is differentiable in $p$ , and $p$ is a local maximum or minimum of $f$ , then $f'(p)=0$ .

So, a local maximum or minimum always is a stationary point.

Please note that as you can see in the first example below, a stationary point is not always a local maximum or minimum.

Let $f$ be a differentiable function. We have seen that if $p$ is a local minimum or maximum of $f$ , it is a stationary point. The opposite is not the case.

Give the function rule $f(x)$ of a function $f$ with stationary point $x=0$ which is not a local extreme.

$f(x)=x^3$

After all, we need to provide a differentiable function $f$ with the following properties:

$f'(0)=0$ (this means that $p=0$ is a stationary point of $f$ )
$0$ is neither a local minimum nor a local maximum of $f$

Because $f'(x)=3x^2$ , we have $f'(0)=0$ , so $0$ is indeed a stationary point of $f$ .

The value of $f$ at $0$ is $0$ . Since $x^3\lt0$ if $x\lt0$ and $x^3\gt 0$ if $x\gt0$ , each interval around $0$ contains points at which the value of $f$ is greater than $f(0)$ and points at which the value of $f$ is smaller than $f(0)$ . In particular, $p=0$ is neither a local minimum nor a local maximum of $f$ .

New example