Question

For three number a, b, c product of the average of the number $a^2$, $b^2$, $c^2$ and $\frac{1}{a^2}$, $\frac{1}{b^2}$, $\frac{1}{c^2}$ can not be less than

• A. 1
• B. 3
• C. 9
• D. none of these

Answer 1

Answer: (A)

1

$\frac{a^2+b^2+c^2}{3}\ge {\left(a^2{b}^2{c}^2\right)}^{1/3}$ using AM$\ge$GM

and $\frac{\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}}{3}\ge {\left(\frac{1}{a^2}\frac{1}{b^2}\frac{1}{c^2}\right)}^{1/3}$

on multiplying both,

$\therefore$ $\left(\frac{a^2+b^2+c^2}{3}\right)\left(\frac{\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}}{3}\right)\ge 1$

$\therefore$ Product of the averages of $a^2$, $b^2$, $c^2$ and $\frac{1}{a^2}$, $\frac{1}{b^2}$, $\frac{1}{c^2}$ can not be less than 1