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I am learning how to find extreme values of functions of two variables. I am getting familiar with critical point concept and I quite don't understand it. I believe I understand what stationary points are, though.

Why one function has no critical points and the other has $P(0,0)$ as its critical point?

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Not sure if that's useful:

  • both functions have no local/global extremas
  • I believe both functions' domains and their derivatives' domains are all equal to each other, and it is $D = (\mathbb{R} \setminus\{ 0\}) \times \mathbb{R}$

Can someone help me understand why one function has critical point and the other does not? How do I find it?

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For the function $f(x,y) = \frac{1}{x} + y$, the partial derivative with respect to $y$, $f_y(x,y) = 1$. This will never equal to $0$ or undefined. Therefore you can't find the critical point. Whereas, for the function $f(x,y) = \frac{1}{x} + y^2$, the partial derivative with respect to $y$, $f_y(x,y) = 2y$ which is equal to $0$ when $y=0$.

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