Maximum value of $\frac{x}{y+1}+\frac{y}{x+1}$ while $0\leq x,y \leq 1$ 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

It is $$\frac{x}{y+1}+\frac{y}{x+1}\le 1$$ since
$$x(x+1)+y(y+1)\le (x+1)(y+1)$$ This is equivalent to
$$0\le x(1-x)+(1-y)(1-x+y)$$
