:eek: Hello I can't do this exercise please help me

Let a, b, c be positive real numbers such that abc=1. Prove that

(a-1+1/b)(b-1+1/c)(c-1+1/a)lessthan or eqal1

Printable View

- Feb 16th 2006, 12:28 AMluckyvanDifficulf exercise! I can't do please help me
:eek: Hello I can't do this exercise please help me

Let a, b, c be positive real numbers such that abc=1. Prove that

(a-1+1/b)(b-1+1/c)(c-1+1/a)lessthan or eqal1 - Feb 16th 2006, 06:51 AMDenMac21Quote:

Originally Posted by**luckyvan**

Since a,b,c are all positive then

must be lower or equal then 3.

It needs proving but I think its easy to prove. I don't have time now but someone else can use this to prove my solution. - Feb 16th 2006, 04:10 PMDenMac21
I have tried to solve it a bit different.

Putting abc instead of 1 (because 1=abc) we get

Now, lets consider next cases:

If a,b,c are all equal to 1 then is true.

If one of them is equal to 1 and others not (for example let be )

we would get:

Knowing that then is equal to:

Because c is positive, (2-c) can have only positive values 1 and 0 and for all other values (2-c) is always negative so

is true.

Hope this solve the problem. - Feb 16th 2006, 04:46 PMtopsquark
It's a pretty little problem, isn't it? It's relatively easy to use the Calculus to show that a=b=1 (where c=1/ab) is an absolute maximum for the expression (and the max = 1 so the expression is less than or equal to 1), but I doubt that this is the approach that the problem was designed for.

-Dan - Feb 16th 2006, 05:50 PMDenMac21
For what aproach is problem designed?

What is the solution?