Question

For what values of x is `f(x)=(x-2)(x-4)(x-3)` concave or convex?

Answer 1

`f(x)` is concave on (-oo,3), and convex on `(3,oo)`.

Explanation:

To find where a function is concave or convex, find where `f''(x)` is positive or negative (respectively).

`f(x)=(x-2)(x-4)(x-3)`

`f'(x)=(1)(x-4)(x-3)+(x-2)(1)(x-3)+(x-2)(x-4)(1)`
`f'(x)=x^2-7x-12+x^2-5x+6+x^2-6x+8`
`f'(x)=3x^2-18x+2`

`f''(x)=6x-18`
`f''(x)=6(x-3)`

`f''(x)` is negative from `(-oo,3)`, and positive from `(3,oo)`

Therefore, `f(x)` is concave on (-oo,3), and convex on `(3,oo)`.