Local maxima, Local minima and saddle points

Question

I am trying to find the local maxima, minima, saddle points and choose the global maxima and minima of this function;

$f(x,y) = e^{-x^2-3y^2}.(x+y^3)$
I tried to find the partial derivatives for x and y. I find the equations and then I use $f_x = 0$ and $f_y = 0$ to find the critical points. However, these equations do not give me points to make a suggestion. Am I doing something wrong?

Answer 1

We have

$$\begin{align} f_x &= e^{-x^2-3 y^2}-2 x e^{-x^2-3 y^2} \left(x+y^3\right) \\ f_y &= 3 y^2 e^{-x^2-3 y^2}-6 y e^{-x^2-3 y^2} \left(x+y^3\right) \end{align}$$

If we simultaneously set $f_x = f_y = 0$, we can divide out $e^{-x^2-3 y^2}$ and we are left with

$$1 - 2x(x+y^3) = 0 \\ 3 y^2 - y(x + y^3) = 0$$

Simultaneously solving

$$(x, y) = \left(\pm \dfrac{1}{\sqrt{2}},0\right)$$