Find points in which grad(f)(x,y) = 0

Question

I need some help with the following task:

Given is $f(x,y) = (4x^2+y^2) \cdot e^{-x^2-4y^2}$ I have to

a) find points $(x_0, y_0)$ for which $\vec \nabla(f)(x_0,y_0) = 0$.

b) calculate eigenvalues and eigenvectors of the hesse matrix of f in those points $(x_0, y_0)$.

c) examine, if the hesse matrix is positive or negative definite.

Here is what I've done so far

1. $\vec \nabla f (x,y) = \begin{pmatrix} -2e^{-x^2-4y^2}\cdot x(4x^2+y^2-4) \\ -2e^{-x^2-4y^2} \cdot y(16x^2+4y^2-1) \end{pmatrix}$

2. critical points:

   • case a: $(x_0, y_0) = (0,0)$.
   • case b: $4x^2+y^2-4 = 0 \wedge 16x^2+4y^2-1 = 0$. There are no real numbers to fulfill this.

-> the teacher said, that there should be 5 points with real entries that could be possible critical points. So: can you tell me what I did wrong?

Answer 1

Hint: There are two more cases: one is $x=0$ and $16x^2+4y^2-1$.

Answer 2

1. Those partial derivatives look wrong to me.
2. You also have the cases (assuming for the moment that your partial derivatives are correct):

$$x = 0, 16x^2+4y^2-1=0$$

$$y = 0, 4x^2+y^2-4=0$$