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If $\frac{\partial f}{\partial x}= 0$ when $x = 1$, and $\frac{\partial f}{\partial x} = (2y-x)$

Do we have to replace $x$ by 1? to get the critical coordinate $y$?

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    Indeed you have :)2012-12-06

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The critical points occur when $f_{x}=f_{y}=0$. So, necessarily, any critical point must occur when $x=1$ so that we obtain $2y-1=0$ and $y=\frac{1}{2}$ as desired. So you are correct.

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    Oh. Sorry. I meant it has the _potential_ to be a critical point. It could be, but we don't know yet. To verify, we must know $f_y$.2012-12-06