Question

If a, b, x, y are positive, then (ab + xy) (ax + by) is

• A. > 2abxy
• B. $\le$ 2abxy
• C. > 4abxy
• D. < 4abxy

Answer 1

The correct option is C

> 4abxy

Since a,b,x,y are positive , ab,xy,ax,by are also positive we know

Arithmetic mean > Geometric mean

$\frac{ab+xy}{2}$ > $\sqrt{\left(ab\right)\left(xy\right)}$

$\Rightarrow$ ab + xy >2 $\sqrt{\left(ab\right)\left(xy\right)}$-------------(1)

Again as AM > GM

$\frac{ax+by}{2}$ >$\sqrt[2]{2axby}$

$\Rightarrow$ ax + by > 2 $\sqrt{abxy}$

From (1) & (2),

(ab + xy) (ax + by) > 4 abxy