Question

If $ a > 0, b > 0, c > 0$, then $(a + b) (b + c) (c + a)$ is greater than

• A. $2 (a + b + c)$
• B. $3 (a + b + c)$
• C. $6\, abc$
• D. $8\, abc$

Answer 1

Answer: (D)

Since A.M. > G.M.

$ \therefore \frac{a+b}{2}> \sqrt{ab}; \frac{b+c}{2} >\sqrt{bc} ; \frac{c+a}{2}> \sqrt{ca} $

$ \therefore \frac{\left(a+b\right)\left(b+c\right)\left(c+a\right)}{8} > \sqrt{ab}\sqrt{bc} \sqrt{ca} = abc $

$\therefore \left(a+b\right)\left(b+c\right)\left(c+a\right) > 8\,abc$