Question

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

Answer 1

`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]