Optimizing a function of 2 variables, necessary conditions

Summary

• $f$ is a real-valued smooth function of two variables on domain $A⊆R^2$ .
• A point $\left( x_0,y_0 \right)∈A$ is called a boundary point if it has an immediate neighboor in $A$ and an immediate neighboor outside $A$ .
• A point $\left( x_0,y_0 \right)∈A$ is called an interior point if all immediate neighboors are in $A$ .
• A point $(x_0,y_0)$ is called a stationary point or a critical point of $f$ if

$$f'_x\left( x_0,y_0 \right)=0 \; \mathrm{and} \; f'_y\left( x_0,y_0 \right)=0$$

• If a stationary point is not a local extreme point it is called a saddle point .
• First derivative test (necessary conditions for extreme values): If a point $(x_0,y_0)$ is a local interior extreme point then $(x_0,y_0)$ must be a stationary point.
• Finding local extreme poits: A point can be a local extreme point of $f$ only if it is a stationary point or a boundary point.