Find all critical values for the function

Question

f(x)=(4x)/(8x^2+8)?

Answer 1

The rational function is f(x) = 4x/(8x²+8).

Differentiate with respect to x.

f'(x) = [4(8x²+8) - 64x²]/(8x²+8)².

f'(x) = [32x² + 32 - 64x²]/(8x²+8)².

f'(x) = 32(1 - x²)/[64(x²+1)²].

f'(x) = (1 - x²)/[32(x²+1)²].

To find the critical numbers f'(x) = 0 and f'(x) does not exist.

f'(x) = 0 ⇒ 1 - x² = 0 ⇒ x = ± 1.

f'(x) does not exist when x²+1 = 0 ⇒ x = ± i.

The imaginary numbers are negligible, so the critical numbers are x = ± 1.