Question

SOMEONE HELP ME WITH THIS PROBLEM PLS!!!!!

Find the maximum(s), minimum(s), interval(s) of increasing, interval(s) of decreasing, end behavior and symmetry for the function f(x)=x^4-8x^2+16.SOMEONE HELP ME WITH THIS PROBLEM PLS!!!!!

Find the maximum(s), minimum(s), interval(s) of increasing, interval(s) of decreasing, end behavior and symmetry for the function f(x)=x^4-8x^2+16.SOMEONE HELP ME WITH THIS PROBLEM PLS!!!!!

Find the maximum(s), minimum(s), interval(s) of increasing, interval(s) of decreasing, end behavior and symmetry for the function f(x)=x^4-8x^2+16.SOMEONE HELP ME

Answer 1

Explanation:

f(x) = x⁴ - 8x² + 16

f'(x) = 4x³ - 16x = 4(x³ - 4x) = 4x(x² - 4) = 4x(x - 2)(x + 2)

f'(x) = 0, x = 2, -2

f"(x) = 4(3x² - 4)

f''(0) = -16 < 0, f""(2) = f''(-2) 3(12 - 4) > 0

so f(2) and f(-2) are minimum values

f(0) is maximum value.

f(0) = 16, f(2) = f(-2) = 2⁴ - 8*2² + 16 = 0

Maximum = 16, minimum value = 0

-∞ < x < -2, f'(x) < 0

-2 < x < 0, f'(x) > 0

0 < x < 2, f'(x) < 0

2 < x < ∞, f(x) > 0

increase: (-2, 0) and (2, ∞)

decrease: (-∞, -2) and (0, 2)