Question

If $a,b,c$ are non-zero real numbers then the minimum value of the expression $\frac{(a^4+a^2+1)(b^4+7b^2+1)(c^4+11c^2+1)}{a^2{b}^2{c}^2}$ equals :-

• A. $315$
• B. $351$
• C. $415$
• D. $451$

Answer 1

Answer: (B)

$351$

We have the given expression,

$\frac{\left(a^4+a^2+1\right)\left(b^4+b^2+1\right)\left(c^4+c^2+1\right)}{a^2{b}^2{c}^2}$

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

$\frac{\left(a^2+\frac{1}{a^2}\right)}{2}⩾\sqrt{a^2+\frac{1}{a^2}}$

$a^2+\frac{1}{a^2}⩾2$

Now, the minimum value is equal to

$=(2+1)(2+7)(2+11)$

$=3\times 9\times 13$

$=351$

Hence the option $\left(B\right)$ is the correct answer.