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Second Derivatives and Beyond

Second Derivatives and Beyond

Topics

Introduction to Second Derivatives And Beyond - At A Glance:

Sample Problem

The maximum value of the function f (x) = -x2 – 1 is y = -1:

Sample Problem

The maximum value of the function f (x) = cos x is y = 1:

Extreme points, also called extrema, are places where a function takes on an extreme value—that is, a value that is especially small or especially large in comparison to other nearby values of the function. Extrema look like the tops of hills and the bottoms of valleys. Time to go hiking.

There are two types of extreme points, minima (the valleys) and maxima (the hills).

Extreme points can be local or global, but we'll talk about this later.

We need to define minimum and maximum values without the on an interval bit.

A minimum value of a function is a y-value of the function that is as low, or lower, than other values of the function nearby. A minimum looks like a valley:

The plural of minimum is minima.

Sample Problem

The minimum value of the function f (x) = x2 + 1 is y = 1:

Sample Problem

The minimum value of the function f (x) = cos x is y = -1:

A function may have multiple minima.

Sample Problem

The function graphed below has two minima: y = 0 and y = 1.

A function may have infinitely many minima.

Sample Problem

The function graphed below has infinitely many minima:

A function may have no minima at all.

Sample Problem

The function f (x) = -x2 has no minima, because for every value of the function there are smaller values nearby:

Be Careful:There is a difference between a minimum of a function (a y-value) and where that minimum occurs (an x-value).

Sample Problem

The minimum value of the function f (x) = x2 + 1 is y = 1, and this minimum occurs at x = 0:

Sample Problem

The function f (x) = cos x has only one minimum value, y = -1. However, this minimum value occurs at infinitely many places, as it occurs at x = π + 2nπ for every integer n:

A function may have multiple maxima.

Sample Problem

The function graphed below has two maxima: y = 2 and y = 3.

A function may have infinitely many maxima.

Sample Problem

The function graphed below has infinitely many maxima:

A function may have no maxima at all.

Sample Problem

The function f (x) = x2 has no maxima, because for every value of the function there are larger values nearby:

Be Careful:There is a difference between a maximum of a function (a y-value) and where that maximum occurs (an x-value).

Sample Problem

The maximum value of the function f (x) = -x– 1 is y = -1, and this maximum occurs at x = 0:

Sample Problem

The function f (x) = cos x has only one maximum value, y = 1. However, this maximum value occurs at infinitely many places, as it occurs at x = 2nπ for every integer n:

Exercise 1

For the function, (a) find all minima of the function, if any, and (b) determine the x-value(s) at which each minima occurs.

Exercise 2

For the function, (a) find all minima of the function, if any, and (b) determine the x-value(s) at which each minima occurs.

Exercise 3

For the function, (a) find all minima of the function, if any, and (b) determine the x-value(s) at which each minima occurs.

Exercise 4

For the function, (a) find all minima of the function, if any, and (b) determine the x-value(s) at which each minima occurs.

Exercise 5

For the function, (a) find all minima of the function, if any, and (b) determine the x-value(s) at which each minima occurs.

Exercise 6

For the function, (a) find all minima of the function, if any, and (b) determine the x-value(s) at which each minima occurs.

f (x) = 2 - x2

Exercise 7

For the function, (a) find all minima of the function, if any, and (b) determine the x-value(s) at which each minima occurs.

f (x) = (x - 4)2 + 3

Exercise 8

For the function, (a) find all minima of the function, if any, and (b) determine the x-value(s) at which each minima occurs.

f (x) = 5

Exercise 9

For the function, (a) find all minima of the function, if any, and (b) determine the x-value(s) at which each minima occurs.

f (x) = 3x + 7

Exercise 10

For the function, (a) find all minima of the function, if any, and (b) determine the x-value(s) at which each minima occurs.

f (x) = sin x

Exercise 11

These questions deal with characteristics of the first and second derivatives of a function at and near a minimum value.

a. Below is a graph of a function f with a minimum at x = x0. Determine the sign of the derivative f' at each labelled x-value.

b. Assume f is defined and twice differentiable on the whole real line. Around a minimum of the function f, is f concave up or concave down?

Exercise 12

These questions deal with characteristics of the first and second derivatives of a function at and near a maximum value.

a. Below is a graph of a function f with a maximum at x = x0. Determine the sign of the derivative f' at each labelled x-value.

b. Assume f is defined and twice differentiable on the whole real line. Around a maximum of the function f, is f concave up or concave down?

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