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.