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# 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

*x*-value(s) at which each minima occurs.

### Example 5

*x*-value(s) at which each minima occurs.

### Example 6

*x*-value(s) at which each minima occurs.

*f* (*x*) = 2 - *x*^{2}

### Example 7

*x*-value(s) at which each minima occurs.

*f* (*x*) = (*x* - 4)^{2} + 3

### Example 8

*x*-value(s) at which each minima occurs.

*f* (*x*) = 5

### Example 9

*x*-value(s) at which each minima occurs.

*f* (*x*) = 3*x* + 7

### Example 10

*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* = x_{0}. 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?

### 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* = *x*_{0}. 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?