What are the critical points of $f(x,y,z)=x^2+ay^2+z^2-4xy, \ \ where \ \ a \in \mathbb{R} $ .

Question

What are the critical points of $ f(x,y,z)=x^2+ay^2+z^2-4xy, \ \ where \ \ a \in \mathbb{R} $ .

Answer:

The critical points are obtained by

$ f_x=0, \\ f_y=0 , \\ f_z=0 \\ $

These gives

$ x-2y=0 , \\ ay-2x=0, \\ z=0 \\ $

This gives $ \ (0,0,0) \ $ only critical points.

Am I right ?

Answer 1

We need to solve the problem of

$$\begin{bmatrix} 1 & -2 \\ -2 & a\end{bmatrix}\begin{bmatrix} x \\ y\end{bmatrix}= \begin{bmatrix} 0 \\ 0\end{bmatrix}$$

What happens if the matrix $\begin{bmatrix} 1 & -2 \\ -2 & a\end{bmatrix}$ is singular? In that case, the origin is not the unique solution.