Question

F(x) = x², [-1, 4] - Find the absolute max and min.

a) Absolute max at x = 4, absolute min at x = -1
b) Absolute max at x = -1, absolute min at x = 4
c) Absolute max at x = 1, absolute min at x = 0
d) Absolute max at x = 0, absolute min at x = 1

Answer 1

The function f(x) = x² has its absolute maximum value at x = 4 and absolute minimum value at x = -1 on the interval [-1, 4], so the correct response is A) Absolute max at x = 4, absolute min at x = -1.

Step-by-step explanation:

The student is asked to find the absolute maximum and minimum of the function f(x) = x² on the interval [-1, 4]. To find the absolute maximum and minimum values of a continuous function on a closed interval, one should evaluate the function at critical points within the interval and the endpoints of the interval. Since the function is a simple quadratic with no critical points other than its vertex, which is outside the interval, we only need to evaluate the endpoints.

By substituting the endpoints into the function, we get:

• f(-1) = (-1)² = 1
• f(4) = (4)² = 16

The absolute maximum value of the function on the interval is at x = 4 and the absolute minimum value is at x = -1, since f(4) > f(-1). Hence the correct option would be:

Provide correct option in final answer:

A) Absolute max at x = 4, absolute min at x = -1.