Calculate local minima and maxima

Question

Consider the function $f : \mathbb{R^2} → \mathbb{R}$ defined by $$f(x_1, x_2)=e^{-(2x_1^2+3x_2^2)}$$ Determine whether the stationary point is a strict local maximum or a strict local minimum. $$$$ So, I found the gradient to be $\nabla{f}=(-4x_1e^{-2x_1^2-3x_2^2}, -6x_2e^{-2x_1^2-3x_2^2})$. Then, $\nabla{f}=0$ in order to find the stationary points and the stationary points are $x_1=0$ and $x_2=0$. But from here I don't know how to proceed anymore. Any help is appreciated.

Answer 1

It is a local max because $e^{-2x_1^2-3x_2^2} \le 1$, and equals to $1$ when $x_1 = x_2 = 0$. It turns out to be a global max because the inequality is true for any ordered pair $(x_1,x_2) \in \mathbb{R^2}$.