Am-Gm inequality

Algebra Level 3

$(a+b)(b+c)(c+a)\geq kabc$ Find the largest positive integer $k$ such that the inequality above holds true for all positive numbers $a,b$ and $c$ .

Bonus : Determine the conditions for equality and prove it.

1 solution

A Former Brilliant Member
Jun 3, 2016

By AM-GM inequality, $a+b \ge 2\sqrt{ab}$ , $b+c \ge 2\sqrt{bc}$ , $c+a \ge 2\sqrt{ca}$ .

Multiplying the 3 inequalities, we get $(a+b)(b+c)(c+a) \ge 8abc$ . Hence $k=8$ .

There is equality if and only if each of three inequalities are equations.

Solving the equations we get equality for $a=b=c$ .

Typo: $a+b \geq 2\sqrt{ab},\;b+c \geq 2\sqrt{bc},\;c+a \geq 2\sqrt{ca}$

Hung Woei Neoh - 5 years ago

Fixed it. Thanks.

A Former Brilliant Member - 5 years ago

Very simple

I Gede Arya Raditya Parameswara - 4 years, 4 months ago