Question

If $a_1,a_2$ and $a_3$ be any positive real numbers, then which of the following statement is true?

• A. $3a_1{a}_2{a}_3\le a_1^3+a_2^3+a_3^3$
• B. $\frac{a_1}{a_2}+\frac{a_2}{a_3}+\frac{a_3}{a_1}\ge 3$
• C. $\left(a_1+a_2+a_3\right){\left(\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}\right)}^{{}^3}\ge 9$
• D. $\left(a_1.a_2.a_3\right){\left(\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}\right)}^{{}^3}\ge 27$

Answer 1

Answer: (D)

We know that, GM $\ge$ HM
$\Rightarrow {\left(a_1.a_2.a_3\right)}^{1/3}\ge \frac{3}{\left(\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}\right)}$
$\Rightarrow \left(a_1.a_2.a_3\right)\ge \frac{27}{{\left(\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}\right)}^{{}^3}}$
$\Rightarrow \left(a_1.a_2.a_3\right){\left(\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}\right)}^{{}^3}\ge 27$