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:  

$$y-\frac{x}{x^2+y^2}=0,x-\frac{y}{x^2+y^2}=0\Rightarrow x^2=y^2\Rightarrow x=\pm y$$
You have to minimize $$xy-\log (x^2+y^2)$$
with given conditions
$$1/4 \le x^2 + y^2 \le 4$$
If $1/4 \le x^2 + y^2 \le 4$ can you find what are the bounds for $x^2$??
If you know what are bounds for $x^2$,
take its minimum value and substitute that in: $xy-\log(x^2+y^2)$
Can you do that now?
