Question

If $p,q,r$ be three distinct real numbers , then the value of $(p+q)(q+r)(r+p)$ is

• A. $>8pqr$
• B. $>16pqr$
• C. $8pqr$
• D. $<8pqr$

Answer 1

Answer: (A)

$>8pqr$

$A.M.=\frac{a+b}{2}$ 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 $a=p$ and $b=q$

$⟹(p+q{)}^2\ge 4pq$ -------(1)

Let $a=q$ and $b=r$

$⟹(q+r{)}^2\ge 4qr$ -------(2)

Let $a=r$ and $b=p$

$⟹(r+p{)}^2\ge 4rp$ -------(3)

multiplying (1) , (2) and (3)

$⟹(p+q{)}^2(q+r{)}^2(r+p{)}^2\ge 64(pqr{)}^2$

applying square root on both sides

$⟹(p+q)(q+r)(r+p)\ge \left(8pqr\right)$