<s>[INST] Solving critical points and classifying the maxima, minima and saddle points I need help to find critical points of the function:
$$f(x,y,z)={(x^2+2y^2+3z^2) e ^{-(x^2+y^2+z^2)}}$$
Then I have to classify these critical points as local maxima/minima or saddle points.
What I would do is to take the partial derivatives $\frac{\partial f}{\partial x}(x, y, z)$, $\frac{\partial f}{\partial y}(x, y, z)$ and $\frac{\partial f}{\partial z}(x, y, z)$. 
But now I will get 
$\frac{\partial f}{\partial x}(x, y, z)$ = $ {2x-6e^{-x^2-y^2-z^2}xz^2} $ 
$\frac{\partial f}{\partial y}(x, y, z)$ = $ {4y-6e^{-x^2-y^2-z^2}yz^2} $ 
$\frac{\partial f}{\partial z}(x, y, z)$ = $ 3(-2e^ {-x^2-y^2-z^2}z^3+2e^{-x^2-y^2-z^2}z) $ 
Normally I would put them in the matrix form and solve the system of equation for x, y and z. 
But I don't know how to solve the system of equation with these partial derivatives. What I am doing wrong or what should be done in order to solve this problem? 
 [/INST] Computing the partial derivatives, we then have
$$\frac{\partial f(x, y, z)}{\partial x}=-2 x  \left(x^2+2 y^2+3 z^2-1\right)e^{-(x^2+y^2+z^2)}$$
$$\frac{\partial f(x, y, z)}{\partial y}=-2 y  \left(x^2+2 y^2+3 z^2-2\right)e^{-(x^2+y^2+z^2)}$$
$$\frac{\partial f(x, y, z)}{\partial z}=-2 z  \left(x^2+2 y^2+3 z^2-3\right)e^{-(x^2+y^2+z^2)}$$ which means that you need to solve
$$x  \left(x^2+2 y^2+3 z^2-1\right)=0$$
$$y  \left(x^2+2 y^2+3 z^2-2\right)=0$$
$$z  \left(x^2+2 y^2+3 z^2-3\right)=0$$ for which the solutions are quite simple
$$\{\{ 0, 0, -1\}, \{0, 0, 1\}, \{0, -1,
   0\}, \{0, 1, 0\}, \{-1, 0, 0\}, \{1, 0,
   0\}, \{0, 0, 0\}\}$$
Now, your turn.
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