Describe the procedure for finding critical points of a function in two independent variables

Describe the procedure for finding critical points of a function in two independent variables.
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Step 1
Based on the theorem on location of extrema, our aim is to
Determine if the exist, and then solve the system
Step 2
$\left\{\begin{array}{l}\frac{\partial }{\partial x}f\left(x,y\right)=0\\ \frac{\partial }{\partial y}f\left(x,y\right)=0\end{array}$
The solution (if any) are the critical points to be further tested.