Question

Suppose f(x) = x^2. Find the absolute extrema of f(x) on each of the following intervals.

A) Interval: [0, 2]
B) Interval: [-1, 1]
C) Interval: [2, 4]
D) Interval: [-2, 0]
Please select the correct option(s).

Answer 1

The absolute extrema of the function f(x) = x^2 on each interval are: [0, 2]: (0, 4), [-1, 1]: (1, 1), [2, 4]: (4, 16), [-2, 0]: (0, 4). Option A.

Step-by-step explanation:

To find the absolute extrema of the function f(x) = x^2 on a given interval, we need to evaluate the function at the endpoints and critical points within the interval.

A) Interval: [0, 2]:

- f(0) = 0^2 = 0

- f(2) = 2^2 = 4

- The absolute minimum is 0 at x = 0 and the absolute maximum is 4 at x = 2.

B) Interval: [-1, 1]:

- f(-1) = (-1)^2 = 1

- f(1) = 1^2 = 1

- The absolute minimum is 1 at x = -1 and the absolute maximum is 1 at x = 1.

C) Interval: [2, 4]:

- f(2) = 2^2 = 4

- f(4) = 4^2 = 16

- The absolute minimum is 4 at x = 2 and the absolute maximum is 16 at x = 4.

D) Interval: [-2, 0]:

- f(-2) = (-2)^2 = 4

- f(0) = 0^2 = 0

- The absolute minimum is 0 at x = 0 and the absolute maximum is 4 at x = -2. Option A.