Easy inequality

Algebra Level 2

If $N$ is the largest positive integer that satisfies the inequality

$a^2c+b^2a+c^2b\geq Nabc,$

where $a,b,c$ are positive reals, find the value of $N$ .

The answer is 3.

4 solutions

Samuraiwarm Tsunayoshi
Jul 19, 2014

Weird solution using rearrangement.

$a^{2}c + b^{2}a + c^{2}b \geq abc+ bca + cab = 3abc$

Which weirdly gives $N = 3$ .

Equality holds iff $a=b=c$ .

Dinesh Chavan
Jul 19, 2014

Simple use of AM-GM yields

$a^2c+b^2a+c^2b \geq 3 \sqrt[3]{a^3b^3c^3}$ Which simply gives $N=3$

Same way, trivial !

Aditya Raut - 6 years, 10 months ago

Level 3??? It doesn't deserve, I suppose. By the way, what the rabbit signifies?

Kartik Sharma - 6 years, 10 months ago

You're genius-that's why you say so!!

Krishna Ar - 6 years, 10 months ago

It is quite simple, don't you think? Well, I think you meant GEN -ius which mean the generation which binds together. Here, bind should mean nuisance. And yes, that's what I am and that's a compliment for me.

Kartik Sharma - 6 years, 10 months ago

@Kartik Sharma – WOW , Nice, GEN-ius Kartik, lol. But if u are intrested in the bunnies story, then u must see this , a friend of mine, Also, u can locate him on fb, I am just like becoming like sree, LOL

Dinesh Chavan - 6 years, 10 months ago

But how can you prove that there can't be a stronger inequality which holds for a larger 'N'.....

Satvik Golechha - 6 years, 10 months ago

Kevin Patel
Jul 22, 2014

General formulatic question, even if you guess with your 6th sense!

Samuel Tan
Jul 21, 2014

Victor, you troller. Rearrange it to get abc+abc + abc = 3abc

Duh.

Easy inequality

The answer is 3.

4 solutions

0 pending reports