Let $f: \mathbb{R}^2 \to \mathbb{R}$ with $f(x, y) = \sin(x + y)\sin(x - y)$. Show that $(x, y) = (0, \frac{\pi}{2})$ satisfies both the necessary and sufficient conditions \to be a local minimizer of $f$.
Hint 1: You will need to take the first and second order derivative of $f$ with respect to $[x, y]$.