Problem working out a second derivative

The second derivative test, that is, computing the second derivative of $f$ does not always tell you whether or not a point is a local maximum or local minimum. It is only always true that if $f''(a) > 0$, then $f$ has a local minimum at $a$.

Hence, when you encounter (as in this case) $f''(a) = 0$, a simple alternative would be to fall back to inspecting the first derivative in an interval around the critical point. For instance, if $x < a \Rightarrow f'(x) < 0$ and $x > a \Rightarrow f'(x) > 0$, then we can safely conclude that the critical point at $x=a$ is a local minimum.

The computations should be trivial given that you have worked out the value of $x = 2\frac{1}{2}$ to be a stationary point.