Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Will the returned result of the function

$$\max\{\tfrac{1}{a}+\tfrac{1}{f}, \tfrac{1}{b}+\tfrac{1}{e}, \tfrac{1}{c}+\tfrac{1}{d}\}$$

return the same set $\{a,f\}$, $\{b,e\}$ or $\{c,d\}$ as the function

$$\min\{a+f, b+e, c+d\}\quad?$$

Assume all numbers are positive and real-valued.

In other words, if, for example, $$a+f < b+e\quad\text{ and }\quad a+f < c+d,$$ will it be true that $$\tfrac{1}{a}+\tfrac{1}{f} > \tfrac{1}{b}+\tfrac{1}{e}\quad\text{ and }\quad \tfrac{1}{a}+\tfrac{1}{f} > \tfrac{1}{c}+\tfrac{1}{d}\quad ?$$

share|cite|improve this question
Do you mean $\text{argmin}$ i.e. the arguments $a,b,\ldots$ that minimize the corresponding sums? – Listing Nov 30 '11 at 9:13
up vote 2 down vote accepted

No. For example, let $a=c=d=f=1$, $b=\frac{1}{2}$, and $e=2$. Then $$\max\{\tfrac{1}{1}+\tfrac{1}{1},\tfrac{1}{2}+\tfrac{1}{\frac{1}{2}},\tfrac{1}{1}+\tfrac{1}{1}\}=\max\{2,\tfrac{5}{2},2\}=\tfrac{5}{2}$$ returns $\{b,e\}$, but $$\min\{1+1,2+\tfrac{1}{2},1+1\}=\min\{2,\tfrac{5}{2},2\}=2$$ returns either $\{a,f\}$ or $\{c,d\}$ (take your pick).

share|cite|improve this answer
Wow, I really screwed up the question. Editing it right now to reflect what I really meant. Your have the right answer for the wrong question (stupid me). – Tabgok Nov 30 '11 at 9:13
@Tabgok: No problem, happens to everyone :) I've now edited my answer to reflect your new question, if I've understood it correctly. – Zev Chonoles Nov 30 '11 at 9:24
Thank you so much! Spent a long time trying to figure this out analytically - looked for a counter-example but everything I tried worked. – Tabgok Nov 30 '11 at 9:43

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.