Show that in any triangle ABC the following inequality holds:cosA+cos B+cos C is less than equal to 3/2 . What is the minimal value of the sum?
We are trying to maximize subject to the constraint .
So
Solve these four equations for .
Spoiler:
So the maximum value is .
For the minimum value, there isn't really one unless we allow and . (Note that this, coupled with , also generates a valid solution to the above system of equations.) Regardless, there is an infimum, and it is .