Question

For what values of x is `f(x)=(x+6)(x-1)(x+3)` concave or convex?

Answer 1

`f(x)` is concave for `x in (-oo, -8/3)` and convex for `x in (-8/3, oo)`

Explanation:

A function is convex where its second derivative is positive and concave where its second derivative is negative. If its second derivative is `0`, then the function may be concave, convex or neither. Commonly such will be a point of inflexion.

Given:

`f(x) = (x+6)(x-1)(x+3)`

`color(white)(f(x)) = x^3+8x^2+9x-18`

We find:

`f'(x) = 3x^2+16x+9`

and:

`f''(x) = 6x+16`

Hence `f(x)` is concave for `x in (-oo, -8/3)` and convex for `x in (-8/3, oo)`

It has a point of infexion at `x=-8/3`

graph{(x+6)(x-1)(x+3) [-8.21, 1.79, -25, 22]}