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

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.

There can be no maximum: $a$ can be made as small as we wish i.e. the expression can be made as large as we wish. — L. F., Oct 04 '13 at 12:14

Luke · Accepted Answer · 2013-10-04T12:13:06.797

0

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.

edited Oct 04 '13 at 12:13

answered Oct 04 '13 at 12:06

Luke

104

As per @L.F.'s comment above, it should probably be mentioned that these methods will work to locate the minimum, but as the constraint is a manifold with boundary which isn't contained in the domain of the function (i.e. not compact), a maximum needn't (and indeed doesn't) exist. – Jonathan Y. Oct 04 '13 at 12:31
@JonathanY. You are correct, thanks for bringing this to my attention. – Luke Oct 04 '13 at 12:45

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.

1 Answers1