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Let $f: M \rightarrow \mathbb{R}$ be a differentiable function that is regular everywhere on the compact manifold with boundary $M$. Show that $f$ assumes its extrema on the boundary.

I know that regular means that the rank of the Jacobian matrix is $1$. Which means that $df_p \neq 0$ for all $p$. Also, since the manifold is compact and $f$ is continuous, it must have a max and a min. But why is it on the boundary? How can I use the fact that $df_p \neq 0$ for all $p$?

Lotte
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    Show Fermat's theorem on the manifold, that a local extremum in the interior would be a point with the differential equal to zero. Do it by contradiction, by looking at points in a small neighborhood and in the direction of the gradient. Use Taylor expansion up to degree $1$. – SphericalTriangle Mar 19 '18 at 18:32
  • @SphericalTriangle, that was so easy. Thanks so much! If you write it as answer, I'll accept it. – Lotte Mar 19 '18 at 18:42
  • You can also write the full answer you have and accept it. There is nothing wrong with that and it is even more productive. – SphericalTriangle Mar 19 '18 at 18:44

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