# Extreme Value Theorem Exercises

### Example 1

Consider the function *f*(*x*) = 2cos(*x*) + 1 on the interval [-2π, 2π]:

- What is the maximum value of the function on this interval?

### Example 2

Consider the function *f*(*x*) = 2cos(*x*) + 1 on the interval [-2π, 2π]:

- What are the values of
*x*at which the maximum is attained?

### Example 3

Consider the function *f*(*x*) = 2cos(*x*) + 1 on the interval [-2π, 2π]:

- What is the minimum value of the function
*f*on this interval?

### Example 4

Consider the function *f*(*x*) = 2cos(*x*) + 1 on the interval [-2π, 2π]:

- What are the values of
*x*at which the minimum is attained?

### Example 5

Let . On which of the following intervals can we use the Extreme Value Theorem to conclude that *f* must attain a maximum and minimum value on that interval?

- (0, π)
- (0, π]
- [0, π]
- (1, 2)
- (1, 2]
- [1, 2]