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?

  • 1
    $\begingroup$ It has no critical points in $\mathbb R^{2}\setminus \{(0,0\}$. $\endgroup$ – Kavi Rama Murthy Nov 21 '18 at 9:53

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