Question

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

Answer 1

`f(x)=(x-3)(x+2)(3x-2)`
`implies f(x)=(x^2-x-6)(3x-2)`
`implies f(x)=3x^3-5x^2-4x+12`

If `f(x)` is a function and `f''(x)` is the second derivative of the function then,

`(i) f(x)` is concave if `f(x)<0`
`(ii) f(x)` is convex if `f(x)>0`

Here `f(x)=3x^3-5x^2-4x+12` is a function.

Let `f'(x)` be the first derivative.
`implies f'(x)=9x^2-10x-4`

Let `f''(x)` be the second derivative.
`implies f''(x)=18x-10`

`f(x)` is concave if `f''(x)<0`
`implies 18x-10<0`
`implies 9x-5<0`
`implies x<5/9`

Hence, `f(x)` is concave for all values belonging to `(-oo,5/9)`

`f(x)` is convex if `f''(x)>0`.
`implies 18x-10>0`
`implies 9x-5>0`
`implies x>5/9`

Hence, `f(x)` is convex for all values belonging to `(5/9,oo)`