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Second Derivatives and Beyond

Second Derivatives and Beyond

Extreme Points and How to Find Them Exercises

Example 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.

Example 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.

Example 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.

Example 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.

Example 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.

Example 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

Example 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

Example 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

Example 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

Example 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

Example 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 ' at each labeled 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?

Example 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 ' at each labeled 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?