If a*b*c=8 then what is minimum value of (2+a)(2+b)(2+c)

Question

If $abc=8$ and $a,b,c >0$, then what is minimum possible value of $(2+a)(2+b)(2+c)$?

Edit: I got the answer and have posted it below.

Answer 1

You can solve this directly using the method of Lagrange multipliers: the critical points are solutions to \begin{align*} (2+a)(2+b) + \lambda a b &=0\\ (2+a)(2+c) + \lambda a c &=0\\ (2+b)(2+c) + \lambda b c &=0\\ abc &= 8, \end{align*} and eliminating equations gives you $a=b=c=2$.

You don't need to consider the case where any of your inequality constraints are active, since your objective function diverges as, say, $a\to0$.

Answer 2

I saw comment of A.M., G.M. inequality and solved it on my own. just posting answer