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# Continuity of Functions

# 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
*f*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]