Question

Locate the absolute extrema of f(x) = ⁴ - ⁵/²x⁴ on [⁰, ²].

a) Absolute maximum at x = ⁰ and absolute minimum at x = ²
b) Absolute maximum at x = ² and absolute minimum at x = ⁰
c) No absolute extrema on the interval [⁰, ²]
d) Absolute maximum at x = ¹ and absolute minimum at x = ²

Answer 1

The absolute maximum is 0 at x = 0 and the absolute minimum is -24 at x = 2.

Step-by-step explanation:

To locate the absolute extrema of the function f(x) = x^4 - 5/2x^4 on the interval [0, 2], we need to find the critical points and endpoints.

Step 1: Take the derivative of f(x) to find the critical points. f'(x) = 4x^3 - 10x^3 = 2x^4 - 14x^3 = 0

Step 2: Solve the equation 2x^4 - 14x^3 = 0 to find the critical points. x(x-7) = 0, so x = 0 or x = 7.

Step 3: Evaluate the function at the critical points and endpoints. f(0) = 0 and f(2) = -24.

Step 4: Compare the values obtained to determine the absolute extrema. The absolute maximum is f(0) = 0 and the absolute minimum is f(2) = -24.