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+4\cdot x+9# and at the point #\rv{2,13}#.

#l(x)= 13#.

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'(2)#. The derivative of #f# is #f'(x)=4-2\cdot x#. This means that #a=f'(2)=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{2,13}#. Hence, #b=13-0 \cdot 2 = 13#. We conclude that the function rule for the tangent is #l(x)=13#.

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