Consider a smooth function $f$ on a compact manifold $M$. Let $p$ be a point where $f$ is maximum and $q$ be a point where $f$ is minimum. Do we necessarily have $\Delta f|_p \leq 0$ and $\Delta f|_q \geq 0$?
2026-04-25 01:19:38.1777079978
Laplacian of a function at points of extrema
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Hint: If a smooth function $f$ of one variable has a local max at $p,$ could $f''(p) > 0?$