Question

For what values of x is `f(x)= 1/2x^4-6x^3-48x^2-96x+2` concave or convex?

Answer 1

`f(x)` is convex when `x` is `(-oo, 2)uu(8, oo)`.

`f(x)` is concave when `x` is `(-2,8)`.

Explanation:

When looking for the concavity of a function, it's best to find the second derivative of the function. The second derivative, `f''(x)`, tells us the concavity of the function, `f(x)`:
When `f''(x)<0`, `f(x)` is concave down (concave)
When `f''(x)>0`, `f(x)` is concave up (convex)

The first derivative of this function is:
`f'(x)` = `2x^3-18x^2-96x-96`.

The second derivative is:
`f''(x)` = `6x^2-36x-96`.

Then set the second derivative equal to 0, to get:
`6x^2-36x-96 = 0`
`x^2-6x-16=0`
`(x-8)(x+2) = 0`
`x= -2, 8`

This tells us that the concavity of `f(x)` changes at `x= -2` and `x= 8`. When `x<-2`, the `f''(x)` is positive. When `x> -2` and `x<8` (i.e. when `x` is between `-2` and `8`) `f''(x)` is negative. When `x>8`, `f''(x)` is positive. It can help to look at a graph:
graph{x^2-6x-16 [-8.67, 11.33, -5.04, 4.96]}

`f(x)` is convex when `f''(x)` is positive, which is when `x` is `(-oo, 2)uu(8, oo)`.

`f(x)` is concave when `f''(x)` is negative, which is when `x` is `(-2,8)`.