For what values of x is f(x)=(2x-2)(x-3)(x+3) concave or convex?

1 Answer
Aug 10, 2018

x in (-oo, 1/3); f(x) is concave and
x in (1/3,oo); f(x) is convex.

Explanation:

f(x)=(2x-2)(x-3)(x+3) or

f(x)= (2x-2)(x^2-9)

f^'(x)= 2(x^2-9)+ (2x-2)*2x or

f^'(x)= 2x^2+4 x^2-4x-18 or

f^'(x)= 6 x^2 -4 x -18

f^''(x)= 12 x -4 ; f^''(x)=0 or 12 x- 4 =0 or x = 1/3

Let’s select a convenient number in the interval less and

more than 1/3 ; x= 0 and 1:. f^"(0)= -4 ; (<0):.

(concave down) and f^''(1)=8 ; (>0):. concave up.

Therefore, x in (-oo, 1/3); f(x) is concave and

x in (1/3,oo); f(x) is convex.

graph{(2x-2)(x-3)(x+3) [-80, 80, -40, 40]} [Ans]