 Solution

At $x=0$, the gradient from the preceding part vanishes and the Hessian matrix is zero. The second-order test is therefore inconclusive. Instead inspect the eigenline:For arbitrarily small positive and negative $t$, these values have opposite signs. Thus $0$ is neither a local maximum nor a local minimum. In dimension at least two it is a degenerate saddle point; in one dimension it is a stationary inflection point. Looking only at the zero Hessian matrix would not determine this classification.