Question

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

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

2021-09-19

Step 1
Based on the theorem on location of extrema, our aim is to
Determine $$\displaystyle{\frac{{\partial}}{{\partial{x}}}}{f{{\left({x},{y}\right)}}}\ \text{ and }\ {\frac{{\partial}}{{\partial{y}}}}{f{{\left({x},{y}\right)}}}$$ if the exist, and then solve the system
Step 2
$$\begin{cases}\frac{\partial}{\partial x} f(x,y)=0\\ \frac{\partial}{\partial y} f(x,y)=0\end{cases}$$
The solution (if any) are the critical points to be further tested.