Question

Find the absolute extrema of the function f(x) = 5 - x on the closed interval [-5, 1].

Answer 1

The absolute maximum of the function f(x) = 5 - x on the closed interval [-5, 1] is 10 at x = -5, and the absolute minimum is 4 at x = 1.

Step-by-step explanation:

To find the absolute extrema of the function f(x) = 5 - x on the closed interval [-5, 1], we analyze the function at the endpoints of the interval and at any critical points within the interval. However, since f(x) is a linear function, it only has one critical point where its derivative is zero. As it's a declining line, we have no critical points within the interval, and the extrema occur at the endpoints. By evaluating the function at the endpoints:

• f(-5) = 5 - (-5) = 10
• f(1) = 5 - 1 = 4

We see that the maximum value of f(x) on the interval is 10 and the minimum value is 4. Thus, the absolute maximum is f(-5) = 10 and the absolute minimum is f(1) = 4.

Answer 2

The absolute maximum of f(x) = 5 - x on the closed interval [-5, 1] is 10, and the absolute minimum is 4.

Step-by-step explanation:

Identify critical points:

The derivative of f(x) is f'(x) = -1.

f'(x) = 0 for no x-value within the closed interval.

Evaluate function at endpoints and critical points:

f(-5) = 10

f(1) = 4

Compare function values:

The maximum value is 10 at x = -5.

The minimum value is 4 at x = 1.

Conclusion:

Absolute maximum: f(-5) = 10

Absolute minimum: f(1) = 4