Optimizing a function of 2 variables, sufficient conditions

Summary

• \(f\) is a real-valued smooth function of two variables on domain \(A⊆R^2\) .
• \((x_0,y_0)∈A_0\) is a stationary point.
• \(H(x_0,y_0)\) is the Hessian of \(f\) evaluated at \((x_0,y_0)\) .
• Second derivative test (sufficient conditions for extreme values):
• If \(H(x_0,y_0)\) is negative definite then \((x_0,y_0)\) is a strict local maximum point
• If \(H(x_0,y_0)\) is positive definite then \((x_0,y_0)\) is a strict local minimum point
• If \(|H(x_0,y_0)|<0\) then \((x_0,y_0)\) is a saddle point
• If \(|H(x_0,y_0)|=0\) then the second derivative test is inconclusive