$$S=\max\left\{{|x-y|\over1+x+y}:0<y<1\right\}\,\,\,\,\forall\,\,\,0<x<1$$
How do I even start this? I can maxmize the numerator by putting $y=0$ for $x\in[1/2,1)$ and $y=1$ for $x\in(0,1/2]$ but since y is in demonimator too they wont necessarily be maxmimas or minimas
Any hints?
We need to find incresing/decresing intervals for $S$
Maximize the ratio over $0<y\leq x$, then maximize over $x<y<1$ and then take the maximum of the two numbers you get. For $0<y\leq x$ the ratio is a decreasing function so its maximum is attained at $y=0$. For $x<y<1$ teh ratio is increasing, so its maximum is attained at $y=1$.