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Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ a function partial differentiable with \begin{align*} \left| \frac{\partial f}{\partial x} \, (x, y) \right| \leq 10 \, \, \, \, \text{and} \, \, \, \left| \frac{\partial f}{\partial y} \, (x, y) \right| \leq 20 \end{align*} for all $(x, y) \in \mathbb{R}^2$ and with $f(0, 0) = 0$. Find the greatest value that $f$ can achieve at the point $(1, 2)$. Give an example of a function $f$ satisfying these assumptions and such that $f(1, 2)$ achieve this value.

I have really no idea to solve this problem (maybe with Taylor's theorem but I don't know how). I only know that a function $f$ satisfying these assumptions is continuous and $L$-lipschitz with $L:=10 \sqrt5$ but I don't see anything else. Can someone help me please? Thanks in advance!

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You can use two line integrals: first along x-axis, then parallel to y-axis. That is,

$$|f(x,y)|\leq f(0,0)+|\int_0^{1}\frac{\partial f}{\partial x} (x, 0) dx|+|\int_0^{2}\frac{\partial f}{\partial y} (1, y) dy|$$ $$\leq 1\cdot 10+2\cdot20=50$$

Conversely $f(x,y)=10x+20y$ makes this bound sharp.

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  • $\begingroup$ Great, thanks a lot! $\endgroup$ – userr777 Feb 4 '18 at 13:19

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