Let $f:{\bf R}^n\to {\bf R}$ ($n\geq 2$) be a $C^1$ function. Is it true that $$\sup_{x\in {\bf R}^n}f(x)=\sup_{x\in {\bf R}^n}f(x+\nabla f(x))\hskip 3pt ?$$

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cannotbe reduced to the almost trivial 1-dimensional case by looking at the line through $x$ and $x+\nabla f(x)$. $\endgroup$ – Yaakov Baruch Nov 29 at 8:18