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## Problem regarding subharmonic functions

Suppose that a function $f: \Bbb{R}^2 \to \Bbb{R}$ of class $C^2$ satisfies the inequality $f_{xx}+f_{yy} \geq 0$ at every point of $\Bbb{R}^2$. Suppose also that all its critical points are non-degenerate, i.e. the matrix of second order derivatives at the critical point has non-zero determinant. Prove that $f$ cannot have local maxima.

Hint: A function with the property that $\Delta f\geq 0$ is called subharmonic. For the subharmonic functions, the maximum principle holds, i.e. the maximum of a subharmonic function cannot be achieved in the interior of the domain. If a subharmonic function has a local maximum then it is locally constant, and therefore has degenerate critical points. This solves our problem.