mathematics also precalculus

Question

Consider the function 𝑓(𝑥) = x² ⁄ x²–1 a) Find the vertical and horizontal asymptotes of the function. b) Find the first derivative of f and determine the maximum and minimum values of f.

and please explain step by step

Answer 1

f(x) = x²/(x²-1) = x²/((x+1)(x-1))

a) The vertical asymptotes are found my setting the denominator to zero and solving for x.

(x+1)(x-1)=0

The horizontal asymptotes are found by taking the lim x→∞.

b)

f '(x) = (2x(x²-1)-x²(2x))/(x²-1)² = (2x³-2x -2x³)/(x²-1)² = -2x/(x²-1)²

The critical points are found by setting f'(x) = 0 and solving for x and where the derivative doesn't exist.

-2x/(x²-1)²=0

-2x=0

x=0

Derivative doesn't exist at (x²-1)²=0 ---> x = ±1

Taking the 2nd derivative test.

f "(x) = 2(3x²+1)/(x²-1)³

f "(0) = -2 <0 then (0,0) is a maximum

f "(±1) undefined