The function is $f(x,y) = \sqrt{x^2+y^2}$ and I need to find its critical points.
The gradient is ($\frac{x}{\sqrt{x^2+y^2}}$, $\frac{y}{\sqrt{x^2+y^2}}$) but the only points where it equals 0 is the point (0,0), where the partial derivatives don't exist.
The calculation of $\lim_{h\to 0}\frac{f(h,0)-f(0,0)}{h}$ yields to: $\lim_{h\to 0}\frac{\sqrt{h^2}}{h} = \lim_{h\to 0}\frac{|h|}{h}$ which doesn't exist.
So what does that means about the critical points?
I know this is late but since there is no answer for this question, I will add one.
The function does have a local/global minima point and its at (0,0) we can confirm this using the graph. But the differentiation method will not work here since the function is not differentiable at (0,0). The function is like a cone with the tip at (0,0)