<s>[INST] 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?
 [/INST] $$ (x^2 + y^2)^{3/2} - y {\frac{3}{2}}(x^2 + y^2)^{\frac{1}{2}}2y = 0$$
$$\iff (x^2 + y^2)^{1/2}\left((x^2 + y^2 - 3y^2)\right) = 0$$
$$\iff (x^2 + y^2)^{1/2}\left(x^2 - 2y^2\right) = 0$$
$$\iff (x^2 + y^2) = 0 \;\;\text{ or }\;\;x^2 = 2y^2$$
$$\iff \quad ?$$
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