Max and Min of $f(x,y)$ Let $f(x,y)=x(y \log y-y)-y \log x$. Find $\smash{\displaystyle\max_{\frac{1}{2} \leq x \leq 2}}(\smash{\displaystyle\min_{\frac{1}{2} \leq y \leq 1} f(x,y)})$. 

EDIT: I have a feeling I'm missing some technique, is there another method I seem unaware of? /EDIT (By the way, from the graph, I'd guess it is at (x,y)=(.5,.5))
First, find the function of $x$, 
$$m(x)=\min_{y\in[.5,1]}f(x,y).$$
Fix $x$ and find $m(x)$ by any means necessary. We could use calculus to find the critical points, but the algebra looks a little tedious. Let's use a graphical program to get a clue: 
http://www.wolframalpha.com/input/?i=plot%28x%28ylogy%E2%88%92y%29%E2%88%92ylogx%2C+.5%3Cx%3C2%2C+.5%3Cy%3C1%29
From the linked plot, if we move in the $y$-direction, at $x=.5$, it is an increasing function, while at $x=2$, it is a decreasing function. Let's just check for critical points after all. We set
$$f_y(x,y)=x\log y-\log x=0,$$ 
so for any $x$, critical points along $y$ occur at they $y$ value $y=x^{1\over x}$. To find the minimum (at fixed $x$), we compare $f(x,x^{1\over x})$, $f(x,.5)$ and $f(x,1)$. 

