Question

On what interval is `f(x)=6x^3+54x-9` concave up and down?

Answer 1

A function is concave up when the second derivative is positive, it is concave down when it is negative, and there could be an inflection point when it is zero.

`y'=18x^2+54`

`y''=36x+54`

so:

`y''>0rArrx> -54/36rArrx> -3/2`.

In `(-3/2,+oo)` the concave is up,

in `(-oo,-3/2)`the concave is down,

in `x=-3/2` there is an inflection point.