Find the inflection points (if any) and the intervals where the function is concave up and concave down for `f(x)=3x^3+x^2+x-9` :

A function is concave up on an interval if the second derivative is positive on the interval; concave down if the second derivative is negative.

`f(x)=3x^3+x^2+x-9`

`f'(x)=9x^2+2x+1`

`f''(x)=18x+2`

`f''(x)=0 ==>x=-1/9`

We try test points on the intervals `(-oo,-2/9),(-2/9,oo)` :

`f''(-1)=-16<0` so the function is concave down on `(-oo,-2/9)`

`f''(0)=2>0` so the function is concave up on `(-2/9,oo)`

Since the concavity changes from down to up at `x=-2/9` and the second derivative exists there` `, `(-2/9,-2237/243)` is the only inflection point.

The graph: