# critical points of $f(x,y) = \sqrt{x^2+y^2}$

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?

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