Question

If abc=8 and a,b,c>0, then the minimum value of (2+a)(2+b)(2+c) IS​

Answer 1

Answer:

What is chapter name what you want to do

Answer 2

given abc=8 solve min. value of (a+2)(b+2)(c+2)

Explanation:

1. referring to the A.M , G.M inequality we know that $A.M\geq G.M$    where A.M- arithmetic mean and G.M- geometric mean
2. for two numbers 'm' and 'n' , $A.M=\frac{m+n}{2}\ \ \ \ \ \ G.M=\sqrt{mn}$  
3. applying this for pairs (a,2) (b,2) and (c,2) we get,                                                            $\frac{a+2}{2} \geq \sqrt{2a} \\\frac{b+2}{2} \geq \sqrt{2b} \\\frac{c+2}{2} \geq \sqrt{2c}$    
4. multiplying these equation we get ,   $\frac{(a+2)(b+2)(c+2)}{8}\geq \sqrt{8abc} \\\frac{(a+2)(b+2)(c+2)}{8}\geq 8\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (given\ abc=8)\\(a+2)(b+2)(c+2)\geq 64$  
5. hence the minimum value is $64$