Problem: Let $f,g : \mathbb{R} ^2 \to \mathbb{R} $ be defined by \[ f(x,y) = x^2 - \frac{3}{2}xy^2 \text{ and } g(x,y) = 4x^4 - 5x^2 y + y^2 \] for all $(x,y) \in \mathbb{R} ^2$. Prove that $(0,0)$ is a critical point for $f$ and $g$. Check the nature of the critical point $(0,0)$ for both these functions.