2 .f(x)=(54((x^2)-1))/((x^2)+27)

Question

Find increasing/decreasing, concavity, all critical points/maximum and minimum /saddle/IPs, and the graph

Answer 1

Simplify f(x)

f(x) = 54(x²-1)/(x²+27)

f(x) = (54x²-54)/(x²+27)

Determine f'(x) using Quotient Rule

f'(x) = [(x²+27)(54x²-54)' - (54x²-54)(x²+27)']/(x²+27)²

f'(x) = [108x(x²+27) - 2x(54x²-54)]/(x²+27)²

f'(x) = (108x³+2916x-108x³+108x)/(x²+27)²

f'(x) = 3024x/(x²+27)²

Determine critical values where f'(x)=0

0 = 3024x/(x²+27)²

0 = 3024x

0 = x

Therefore, the only critical value of f(x) is at x=0.

Use test points to see where f(x) increases and decreases

f'(-1) = 3024(-1)/((-1)²+27)² = -3024/(28²) = -3024/784 < 0

f'(1) = 3024(1)/(1²+27)² = 3024/(28²) = 3024/784 > 0

Thus, f(x) decreases over the interval (-∞,0) and increases over the interval (0,∞), which makes x=0 a local minimum.

Find possible inflection points (IPs) where f''(x)=0

f'(x) = 3024x/(x²+27)²

f''(x) = [(x²+27)²(3024x)' - (3024x)((x²+27)²)']/((x²+27)²)²

f''(x) = [3024(x²+27)² - (3024x)(4x(x²+27))]/(x²+27)⁴

f''(x) = [3024(x²+27)² - 12096x²(x²+27)]/(x²+27)⁴

f''(x) = [3024(x²+27) - 12096x²]/[(x²+27)³

f''(x) = (3024x² + 81648 - 12096x²)/(x²+27)³

f''(x) = (81648-9072x²)/(x²+27)³

0 = (81648-9072x²)/(x²+27)³

0 = 81648-9072x²

9072x² = 81648

x² = 9

x = ±3

Therefore, (-3,12) and (3,12) are inflection points of f(x)

To get the graph, use the Desmos graphing calculator and insert f(x) = (54x²-54)/(x²+27)

Hope this helped!