<s>[INST] max and min $x y - \ln(x^2 + y^2)$ Find max and min
$$x y - \ln(x^2 + y^2) , 1/4 \le x^2 + y^2 \le 4$$
Problem: With this hairy expression as my partial derivatives, I do not know what to do.
Attempt:  
 [/INST] \begin{align}
x-\frac{2y}{x^2+y^2}=0\\
y-\frac{2x}{x^2+y^2}=0
\end{align}
can be combined to, as you have done, to
\begin{align}
(x+y)\left(1-\frac2{x^2+y^2}\right)=0\\
(x-y)\left(1+\frac2{x^2+y^2}\right)=0
\end{align}
and also, since $(x,y)\ne(0,0)$, to
\begin{align}
2xy-2=0
\end{align}
</s>