<s>[INST] max and min of $f(x,y)=\frac{xy+y^2}{x^2+y^2}$ The function is symmetric because $f(x,y)=f(-x,-y)$. It's defined in $\mathbb{R}^2-\{{(0,0)}\}$.If I calculate the gradient , it's null in the origin. Have I to prove $f$ is limited in $\mathbb{R}^2$?
 [/INST] By the arithmetic mean-geometric mean inequality
$$
    |xy| \leq \tfrac{1}{2}(x^2 + y^2)
$$
so
$$
    \left|\frac{xy + y^2}{x^2 + y^2}\right| \leq \frac{|xy|}{x^2 + y^2} + \frac{|y|^2}{x^2 + y^2} \leq \frac{1}{2} + 1 = \frac{3}{2}
$$
Since $|f(x,y)| \leq \frac{3}{2}$, we know
$$
    -\frac{3}{2} \leq f(x,y) \leq \frac{3}{2}
$$
and $f$ is bounded both above and below.
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