Question

Show $:(ab+xy)(ax+by)\ge 4abxy$ where $a,b,x,y\in R.$

Answer 1

Since $a,b,x,y$ are positive numbers
$ax,by,ab,xy$ are also positive

Now $\because AM\ge G.M$
$\Rightarrow \frac{ax+by}{2}\ge \sqrt{axby}$
$\Rightarrow ax+by\ge 2\sqrt{abxy}$ ...(1)

Also $\frac{ab+xy}{2}\ge \sqrt{abxy}$
$\Rightarrow ab+xy\ge 2\sqrt{abxy}$ ...(2)

From (1) and (2)
$(ax+by)(ab+xy)\ge 4abxy$