Question

If a>0, b>0, c>0 and the minimum value of $a(b^2+c^2)+b(c^2+a^2)+c(a^2+b^2)$ is $\lambda abc$ ,then $\lambda$ is

• A. 2
• B. 1
• C. 6
• D. 3

Answer 1

The correct option is C 6

Taking the numbers $a{b}^2$ and $a{c}^2$
$\frac{a{b}^2+a{c}^2}{2}\ge \sqrt{a{b}^2.a{c}^2}$
$\Rightarrow a{b}^2+a{c}^2\ge 2abc$ ________(1)
Similarly, $b{c}^2+b{a}^2\ge 2abc$ _______(2)
and $c{a}^2+c{b}^2\ge 2abc$ ________(3)
Adding $\left(1\right),\left(2\right)$ and $\left(3\right)$
$a(b^2+c^2)+b(c^2+a^2)+c(a^2+b^2)\ge 6abc$
Hence, option C is correct