TITLE: How to find the maximum of $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{a+b}+\frac{1}{a+c}$ given certain constraints.
QUESTION [3 upvotes]: Let $a,b,c\ge 0,$ and such $a+b+c=1$. Find the maximum of:
$$\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+\dfrac{1}{a+b}+\dfrac{1}{a+c}$$
My try: 
$$\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+\dfrac{1}{a+b}+\dfrac{1}{a+c}=\dfrac{1}{1-b-c}+\dfrac{1}{b}+\dfrac{1}{c}+\dfrac{1}{1-c}+\dfrac{1}{1-b}$$
Then I can't, thank you.

REPLY [0 votes]: Hint:

Kuhn-Tucker Multipliers For inequality constraints
Lagrange Multipliers For equality constraints

Another Hint:
The constraint $a+b+c=1$ makes a variable redundant. For example, $a$ may be substituted with $a = 1-b-c$ (which you have correctly identified), so only the inequality constraints need to be considered.
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