If a, b, c are the positive integers, then (a + b)(b + c)(c + a) is 

Concept:

If a,b,c are the positive integers, then

1. Arithmetic mean:

  \({\rm{A}} = \frac{{{\rm{a\;}} + {\rm{\;b}}}}{2}\)

2. Geometric mean 

  \({\rm{G}} = \sqrt {{\rm{ab}}} \)

3. Harmonic mean 

  \({\rm{H}} =\ \frac{2}{\frac{1}{a}\ +\ \frac{1}{b}} =\ \frac{{2{\rm{ab}}}}{{{\rm{a}} + {\rm{b}}}}\)

4. The relation between AM, GM, and HM

• G2 = AH
• AM  ≥  GM  ≥  HM

Calculation:

We know that, 

AM ≥ GM

⇒ \(\frac{a+b}{2}≥ \sqrt{ab}\)   ---(1)

Similarly,

\(\frac{b+c}{2}≥ \sqrt{bc}\)     ----(2)

\(\frac{c+a}{2}≥ \sqrt{ca}\)   ----(3)

Multiplying these inequalities, we will get

\(\frac{(a+b)(b+c)(c+a)}{2\times 2\times2} ≥\sqrt{a^2b^2c^2}\)

⇒  (a + b)(b + c)(c + a)  ≥ 8abc

Hence, ≥ 8abc is the correct answer.