Question

How do you determine whether the function `f(x) = 2x^3 + 3x^2 - 432x` is concave up or concave down and its intervals?

Answer 1

f(x) is concave down on `(-oo;-1/2)` and concave up on `(-1/2;oo)`

Explanation:

You have to investigate the sign of the second derivative around its zero points (inflection points).

In this case,

`f'(x)=6x^2+6x-432`
`f''(x)=12x+6`

`therefore f''(x)=0iffx=-1/2`

`therefore AAx<-1/2=>f''(x)<0` and hence the concavity is down.

`therefore AAx >-1/2=>f''(x)>0` and hence the concavity is up.