<s>[INST] Monotonicity of $f(x,y)=\frac{x^2+y^2}{x+y}$ $f(x,y)=\frac{x^2+y^2}{x+y} $ where $x,y \in [0,1]$
let $x\leq r$ and $y\leq s$ I have to prove\disprove $f(x,y)\leq f(r,s)$. 
Clearly $x^2+y^2\leq r^2+s^2$ and $x+y\leq r+s$ but then $\frac{1}{x+y}\geq \frac{1}{r+s}$ ??
 [/INST] It is not true.  
As one of many counterexamples, consider $x=r=1$ and $y=0, s=\frac12$
Then $\dfrac{1^2+0^2}{1+0}=1 \gt \dfrac56 = \dfrac{1^2+\left(\frac12\right)^2}{1+\frac12}$ so $f(x,y) \not \le f(r,s)$ even though $x \le r$ and $y \le s$
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