Consider the function f(x) = 2cos(x) + 1 on the interval [-2π, 2π]:
What is the maximum value of the function on this interval?
Answer
The maximum value of the function on this interval is 3.
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?
Answer
f(x) = 3 when x = -2π, x = 0, or x = 2π.
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?
Answer
The minimum value of the function on this interval is -1.
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?
Answer
f(x) = -1 when x = -π, or x = π.
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]
Answer
(0, π): We can't use the Extreme Value Theorem because this interval is not closed.
(0, π]: We can't use the Extreme Value Theorem because this interval is not closed.
[0, π] - We can't use the Extreme Value Theorem because f is discontinuous at x = 0, so f is not continuous on this interval.
(1, 2): We can't use the Extreme Value Theorem because this interval is not closed.
(1, 2]: We can't use the Extreme Value Theorem because this interval is not closed.
[1, 2]: We can use the Extreme Value Theorem because this interval is closed and f is continuous on this interval.
. On which of the following intervals can we use the Extreme Value Theorem to conclude that f must attain a maximum and minimum value