Question

On [0,1], let f have continuous derivatives satisfying 0<ƒ'(t)≤1. Also suppose ƒ(0)=0. Prove that (int_{0}^{1}ƒ(t)dt)^{2}≥int_{0}^{1}(ƒ(t))^{3}dt

Answer 1

\[(\int\limits_{0}^{1}ƒ(t)dt)^{2}≥\int\limits_{0}^{1}(ƒ(t))^{3}dt \]

Answer 2

that's what it looks like

Answer 3

This is just the integral version of the Cauchy-Schwarz inequality.

Answer 4

This guy/girl is actually doing CS inequality without actually knowing it.

Answer 5

Mhm, just CS in the limit. Now that you know the name, you can probably find a dozen proofs of this fact on the web (probably even wikipedia).

Answer 6

Yesterday,he/she wanted was asking for the proof of the standard version.

Answer 7

Ah. I don't know what proof you gave him, but my favorite proof is just \[\frac{u \cdot v}{|u||v|}=\cos \theta\] where dot product is just taken to be the appropriate metric. For the integral case, I believe it is just the standard integral metric.

Answer 8

I just pointed him/her to the Wikipedia page (I guess!)

Answer 9

Btw this is probably the most useful inequality in competitive mathematics so much application.

Answer 10

Well, I am sure you are aware of this :)

Answer 11

It most definitely is! That and the rearrangement inequality. I had never really understood how powerful that technique was until this past year, where our coach made us prove an entire sheet of problems using two different methods, one of which had to be rearrangement!

Answer 12

Cool, have you got your hands on "The CS master class" ?

Answer 13

Excellent book! I have mine loaned out to a friend right now (Putnam is over, so no more competition and all, haha!), but that is definitely one of the best texts on competition inequalities. That and Hoojoo Lee's treatise.

Answer 14

Oh how I wish I knew what you guys were talking about. All I know is the basic idea of CS. Where is a good source from which I can learn more about it's applications?

Answer 15

I envy you man, I never had the opportunity to lay my hands on it :( The imported copy is pretty expensive.

Answer 16

Master class Turing that's the best.

Answer 17

Lee's treatise can be found here: http://www.eleves.ens.fr/home/kortchem/olympiades/Cours/Inegalites/tin2006.pdf

Basically, every imo competitor has that completely memorized back to front.

And Master Class, by Steele I believe, is a great book on the subject. I'm fortunately in the US, so it's not too hard to come by books like that.

Answer 18

well I'm not IMO material, but thanks for the source, I'll see how far I can go with it. I really only want to use it for physics.

Answer 19

I also mostly learned it from various sources until college at least. But the past two years I've had a really good coach and have picked up more of an appreciation for reading texts like that. Hardy/Littlewood/Polya book Inequalities is also pretty good (more for academics than competitions) and is ancient enough you can probably find it on the web for free.