We have, $z=ax+by,a,b>0$
Subject to constraints $x\le 2,y\le 2,x+y\ge 3x,y\ge 0$
On taking given constraints as equation, we get the following graph

Here, ABCA is the required feasible region whose corner points are $A\left(\mathrm{2,1}\right),B\left(\mathrm{1,2}\right)$ and $C\left(\mathrm{2,2}\right)$ .
Since, It is given that $z=ax+by;a,b>0$
has minimum value at $\left(\mathrm{2,1}\right)$
$\therefore$ Value of z at $\left(\mathrm{2,1}\right)<$ value of z at $\left(\mathrm{1,2}\right)$
$\Rightarrow 2a+b<a+2b$
$\Rightarrow a<b$