Set f'(x) = 0. The values of x where this happens are your candidates.
So f'(x) = 0 only when x = 0.
Now is this a relative max or a relative min? You can find out by graphing, but we need the second derivative to get the other stuff anyway, so let's do the second derivative test on this.
(after a bit of factoring)
So what is f''(x) for x = 0? . Since f''(0) is negative, the point (0, -4/9) is a relative maximum.
As it happens, the second derivative IS the concavity, so you have already found this.
An inflection point is where f''(x) = 0. So this happens when , which has no real solutions for x. Thus there are no inflection points.