What is the maximum value of $\frac{x}{y+1}+\frac{y}{x+1}$ while $0\leq x,y \leq 1$?
Wolfram Alpha plots this expression on a 3d graph, but I want to solve it algebraicly, by modifying the expression
My Attempts
1) add and substract 2 at the equation and we get $\frac{x+y+1}{y+1}+\frac{x+y+1}{x+1}$ and the numerator is same
=>failed
2) use AM-GM or Cauchy-Schwarz inequality
=>also failed
Maximum value of $\frac{x}{y+1}+\frac{y}{x+1}$ while $0\leq x,y \leq 1$
a.m.-g.m.-inequalitycauchy-schwarz-inequalityinequality
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Best Answer
When $ 0\le x,y \le 1$, then $$\frac{x}{y+1} \le \frac{x}{x+y}$$ $$\frac{y}{x+1} \le \frac{y}{x+y}$$ Adding them we have $$\frac{x}{y+1}+\frac{y}{x+1} \le \frac{x+y}{x+y}=1.$$
Equality holds when $x=0$ and $y=1$ or $x=1$ and $y=0$ and the maximum of $1$ is attained.