Question

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

• A. $3a_{1}a_{2}a_{3}\le a^{3}_{1}+a^{3}_{2}+a^{3}_{3}$
• B. $\frac{a_{1}}{a_{2}}+\frac{a_{2}}{a_{3}}+\frac{a_{3}}{a_{1}}\ge3$
• C. $\left(a_{1}+a_{2}+a_{3}\right) \left(\frac{1}{a_{1}}+\frac{1}{a_{2}}+\frac{1}{a_{3}}\right)^{^3} \ge9$
• D. $\left(a_{1}.a_{2}.a_{3}\right) \left(\frac{1}{a_{1}}+\frac{1}{a_{2}}+\frac{1}{a_{3}}\right)^{^3} \ge27$

Answer 1

Answer: (D)

We know that, GM $\ge$ HM
$\Rightarrow \left(a_{1}.a_{2}.a_{3}\right)^{1/3}\ge\frac{3}{\left(\frac{1}{a_{1}}+\frac{1}{a_{2}}+\frac{1}{a_{3}}\right)}$
$\Rightarrow \left(a_{1}.a_{2}.a_{3}\right)\ge\frac{27}{\left(\frac{1}{a_{1}}+\frac{1}{a_{2}}+\frac{1}{a_{3}}\right)^{^3}}$
$\Rightarrow \left(a_{1}.a_{2}.a_{3}\right) \left(\frac{1}{a_{1}}+\frac{1}{a_{2}}+\frac{1}{a_{3}}\right)^{^3} \ge27$