Question

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

• A. $1$
• B. $3$
• C. $9$
• D. None of these

Answer 1

Answer: (A)

$\frac{a^2+b^2+c^2}{3}\ge {\left(a^2{b}^2{c}^2\right)}^{1/3}$ using A.M. $\ge$ 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
$\therefore \left(\frac{a^2+b^2+c^2}{3}\right)\cdot \left(\frac{\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}}{3}\right)\ge 1$
$\therefore$ Product of the averages of $a^2,b^2,c^2$ and $\frac{1}{a^2},\frac{1}{b^2},\frac{1}{c^2}$ cannot be less than $1$