Question

For what values of x is `f(x)=(x+6)(x-10)(x-3)` concave or convex?

Answer 1

`f’’(x) < 0` at `x = [-oo,1.167]` (concave)
`f’’(x) > 0` at `x = [1.17, +oo]` (convex)

Explanation:

First we combine the terms into a singe polynomial. Then we take the first and second derivatives. Where f’(x) is 0 are the local maxima and/or minima.
If f”(x) > 0 it is convex, if f”(x) < 0 it is concave. `f(x) = (x + 6)(x − 10)(x − 3) = (x^2 -4x -60)(x – 3)`
`= x^3 – 4x^2 – 60x -3x^2 + 12x -180 = x^3 – 7x^2 – 48x – 180`

`f(x) = x^3 – 7x^2 – 48x – 180` ; `f’(x) = 3x^2 – 7x – 48` ; `f’’(x) = 6x – 7` The equation is a cubic, so there is only an inflection point, but no minima or maxima.
`f’’(x) < 0` at `x = [-oo,1.167]` (concave)
`f’’(x) > 0` at `x = [1.17, +oo]` (convex)