Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I'm having a really hard time knowing where to start with this, I'd appreciate if anyone could give me a hand!

Prove or disprove the following statement. The particular function that extremises a certain functional J among all the functions that render l to another functional K also gives an extremum to the functional K among all the functions that give J a precribed value.

Thanks

share|improve this question
    
As in: The unit square has minimal circumference among all rectangles of area $1$ and also maximal area among rectangles of circumference $4$. –  Hagen von Eitzen Nov 22 '12 at 18:54

1 Answer 1

Assume $g$ has the property that $K(g)=1$ and that $K(f)=1$ implies $J(g)\ge J(f)$.

Let $c=J(g)$. If $f$ is any function with $J(f)=c$, let $y=K(f)$. Then $K(\frac1yf)=1$, hence $\frac cy=\frac1y J(f)=J(\frac1yf)\le J(g)=c$. We would like this to imply $y\ge 1$, however there are a few obstacles:

  • What if $c<0$? No problem, we'd find $<\le 1$ instead - provided the other obstacles don't apply.
  • What if $y<0$?
  • In fact, what if $y=0$ and we mustn't divide by $y$?
  • And what if $c=0$?
share|improve this answer

Your Answer

 
discard

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.