Solving an equation with fractional powers I was trying to find the maximum value for a function. I took the first derivative and arrived at this horrible expression:
$$ (x^2 + y^2)^\frac{3}{2} - y {\frac{3}{2}}(x^2 + y^2)^{\frac{1}{2}}2y = 0$$
How can I find the extrema by hand?

After some simplifying steps,
$$(x^2 + y^2)^\frac{3}{2} - y {\frac{3}{2}}(x^2 + y^2)^{\frac{1}{2}}2y = 0, \\
(x^2 + y^2)^\frac{3}{2} - 3y^2(x^2 + y^2)^{\frac{1}{2}} = 0, \\
(x^2 + y^2)^{\frac{1}{2}}\left(x^2+y^2-3y^2 \right)=0, \\
x^2+y^2=0 \Rightarrow x=0,\;y=0, \\
\text{or}\\
x^2-2y^2=0 \Rightarrow |x|=\sqrt{2}|y|.
$$
