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