Find the absolute maximum and minimum values, and their locations, of the function f(x) in the given interval f(x) = x²−1, −1 ≤ x ≤ 2.

The highest value of the function in the domain is considered the maximum value of a function, and the lowest value is considered the minimum value of the function.

Answer: The absolute maximum and minimum values of the given function f(x) = x²−1, −1 ≤ x ≤ 2, are 3 and -1 respectively and they occur at x = 2 and x = 0, respectively.

Let us see how to find absolute maximum and minimum

Explanation:

To find the absolute maximum and minimum, we will determine the derivative of the function f(x) = x² − 1,

f'(x) = 2x

Put f'(x) = 0 ⇒ 2x = 0 ⇒ x = 0

Hence, x = 0 is a critical point.

Now, we will check the values of f(x) for x = -1, 0, 2 (Critical points and and end points of the domain)

f(-1) = (-1)² - 1 = 0

f(0) = 0² -1 = -1 

f(2) = 2² - 1 = 3

Hence, from above we can see that the absolute minimum value of the function f(x) is -1 and it occurs at x = 0 and the absolute maximum value is 3 and it occurs at x = 2.

Thus, the absolute maximum and minimum values of the given function are 3 and -1 respectively and they occur at x = 2 and x = 0, respectively.