Question

$P=(ab+xy)(ax+by)$, then:

• A. $P>10abxy$
• B. $P>16abxy$
• C. $P>4abxy$
• D. $P>2abxy$

Answer 1

Answer: (C), (D)

$P>4abxy$
$P>2abxy$

$A.M.=\frac{a+b}{2}$ and $G.M.=\sqrt{ab}$

Subtracting above equations we get

$A.M.-G.M.=\frac{a+b}{2}-\sqrt{ab}$

$=\frac{a+b-2\sqrt{ab}}{2}$ ------- (1)

$=\frac{(\sqrt{a}-\sqrt{b}{)}^2}{2}\ge 0$ (since, square term is always $\ge 0$)

$⟹A.M.-G.M.\ge 0$

Therefore, $A.M.\ge G.M.$

$⟹\frac{a+b}{2}\ge \sqrt{ab}$

squaring on both sides.

$⟹(\frac{a+b}{2}{)}^2\ge ab$

$⟹(a+b{)}^2\ge 4ab$

Let $a=ab$ and $b=xy$

$⟹(ab+xy{)}^2\ge 4abxy$ -------(1)

Let $a=ax$ and $b=by$

$⟹(ax+by{)}^2\ge 4abxy$ -------(2)

multiplying (1) and (2)

$⟹(ab+xy{)}^2(ax+by{)}^2\ge (4abxy{)}^2$

applying square root on both sides

$⟹(ab+xy)(ax+by)\ge \left(4abxy\right)$

Let $a=2ab$ and $b=2xy$

$⟹(2ab+2xy{)}^2\ge 4abxy$

$⟹(ab+xy{)}^2\ge abxy$ -------(3)

multiplying (3) and (2)

$⟹(ab+xy{)}^2(ax+by{)}^2\ge 4(abxy{)}^2$

applying square root on both sides

$⟹(ab+xy)(ax+by)\ge \left(2abxy\right)$