+170 Let \(a,b,c\) be nonnegative real numbers such that \(a + b + c = 1.\) Find the maximum value of \(\frac{ab}{a + b} + \frac{ac}{a + c} + \frac{bc}{b + c}.\)
+6251 In all these symmetric problems like this, with non-negative elements, either all the elements are equal or all but one element is zero.
We just have to figure out which corresponds to the min and which to the max.
If a,b,c=1/3, each term in the sum is 1/6, so the sum is 1/2
if a=1, b=c=0, oops can't do that, we need at least 2 non-zero terms.
Suppose a=b=1/2, c=0
1/4 + 0+0 = 1/4 < 1/2
So the maximum occurs when all the variables are equal at 1/3 and is equal to 1/2.
You can show this formally using Lagrange Multipliers if you like.
I'm not sure what method they intend for you to use.
I suspect they want you to unmangle the expression so that the Cauchy-Schwarz inequality can be applied.
