An interesting inequality

[Deuteragonist]

Given a+b+c=3, prove that:
(ab+c)/(a+b)+(bc+a)/(b+c)+(ca+b)/(c+a) is greater than or equal to 11/4.

Any help would be much appreciated.

 

[MarkFL]

I would begin by observing we have cyclic symmetry, that is, exchanging any two of the 3 variables does not change the objective function or the constraint, and so we know a critical point is:

[MATH](a,b,c)=(1,1,1)[/MATH]
What is the value of the objective function at that point. What is its value at some other point on the constraint, and how does it compare to the value at the critical point?

 

[Steven G]

I would begin by observing we have cyclic symmetry, that is, exchanging any two of the 3 variables does not change the objective function or the constraint, and so we know a critical point is:

[MATH](a,b,c)=(1,1,1)[/MATH]
What is the value of the objective function at that point. What is its value at some other point on the constraint, and how does it compare to the value at the critical point?

Mark, I see the symmetry you are speaking about (I even saw it before I saw your post) but I do not know why you call (1,1,1) (an obvious point in the solution) a critical point (and yes I do NOT think you are referring to calculus). Can you please enlighten me. Thanks!!!!

 

[MarkFL]

With cyclic symmetry, we let:

[MATH]a=b=c[/MATH]
And from the constraint, this means:

[MATH]a=b=c=1[/MATH]