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.
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.