approximating a maximum function by a differentiable function Is it possible to approximate the $max\{x,y\}$ by a differentiable function?
$f(x,y)=max \{x,y\} ;\ x,y>0$

Yes it is. One possibility is the following: Note that $\def\abs#1{\left|#1\right|}$
$$ \max\{x,y\} = \frac 12 \bigl( x+ y + \abs{x-y}\bigr), $$
take a differentiable approximation of $\abs\cdot$, for example $\def\abe{\mathop{\rm abs}\nolimits_\epsilon}$$\abe \colon \mathbb R \to \mathbb R$ for $\epsilon > 0$ given by
$$ \abe(x) := \sqrt{x^2 + \epsilon}, \quad x \in \mathbb R $$
and define $\max_\epsilon \colon \mathbb R^2 \to \mathbb R$ by 
$$ \max\nolimits_\epsilon(x,y) := \frac 12 \bigl( x+y+\abe(x-y)\bigr). $$
Another possibility is to take a smooth mollifier $\phi_\epsilon$ and let $\max'_\epsilon :=\mathord\max * \phi_\epsilon$.
