<s>[INST] Finding minimum $\frac{x+y}{z}+\frac{x+z}{y}+\frac{y+z}{x}$ I would appreciate if somebody could help me with the following problem
Q. Finding maximum minimum 
$$\frac{x+y}{z}+\frac{x+z}{y}+\frac{y+z}{x}(\text{where} ~~x,y,z>0)$$
 [/INST] $$\frac{x+y}{z}+\frac{x+z}{y}+\frac{y+z}{x}(\text{where} ~~x,y,z>0)$$
Consider the following : 
$a , \frac{1}{a}$ we know that $A.M. \geq G.M.$
$\therefore \frac{a+ \frac{1}{a}}{2} \geq \sqrt{a . \frac{1}{a}}$
$\Rightarrow \frac{a^2+1}{2a} \geq 1 $
$\Rightarrow a^2 + 1 \geq 2a $
$\Rightarrow (a-1)^2 \geq 0$
$\Rightarrow a \geq 1$ 
$\therefore $ the expression has only minimum value which is 1 and no maximum value. 
The expression $$\frac{x+y}{z}+\frac{x+z}{y}+\frac{y+z}{x}(\text{where} ~~x,y,z>0)$$
has minimum value of 1 + 1 +1 +1 +1 +1+1 = 6 
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