Question

Fun question

If f(0) = f(1) = 0 and f" exists, show that

Answer 1

\[\int\limits_{0}^{1} f''(x)f(x)dx = - \int\limits_{0}^{1} [f'(x)]^2 dx\] shouldn't be too bad :)

Answer 2

LOL

Answer 3

Alright what if you combine \(f''(x)f(x) \implies (f'''(x)dx)\) then do the chain rule twice?

Answer 4

something like that.

Answer 5

Hmm.

Answer 6

My initial thought:\[
(ff')' = f'f'+ff''
\]

Answer 7

That's where I was going with it.

Answer 8

Guess triple prime wasn't needed.

Answer 9

by parts !

Answer 10

wio U are right

Answer 11

(ff')=(f')^2+ff"

Answer 12

\[\int\limits_{0}^{1} f''(x)f(x)dx = f'(x)f(x)\Bigg|_{0}^1 - \int \limits_0^1 f'(x) f'(x) dx\]

Answer 13

We know that: \[
\int_0^2 (f(x)f'(x))'dx=\int_0^1d\bigg(f(x)f'(x)\bigg)= 0f'(2)-0f'(0)= 0
\]

Answer 14

And oo:\[\int\limits_{0}^{1} f''(x)f(x)dx +\int\limits_{0}^{1} [f'(x)]^2 dx = 0\]

Answer 15

and so

Answer 16

@iambatman Is this enough?

Answer 17

Yes, that's enough haha, maybe next time I'll have a no ganeshie rule! :P

Answer 18

What? I came up with the anti-derivative first thought...

Answer 19

though

Laptop keyboard is unreliable

Answer 20

lol ask tough questions instead

Answer 21

shots fired

Answer 22

It wasn't suppose to be tough :(, just fun

Answer 23

pew pew

Answer 24

Guess, I'll find some questions related to eliassaab's haha, if you want a real challenge :P

Answer 25

I think we exceeded the quota for number theory this week, im eager to do more calc
post more questions batman xD

Answer 26

Haha, well mhm, I'll have to find/ think of some tricky ones for calc...I will see :D, and I agree there have been many number theory questions the past week D:

Answer 27

does\( f(x)=xsin(\pi x)\) satisfies it

Answer 28

is it twice differentiable ?

Answer 29

http://www.wolframalpha.com/input/?i=%28xsin%28pi+x%29%29%27%27

Answer 30

yes i think

Answer 31

yes it satisfies the hypothesis so both integrals need to evaluate to same numbers

Answer 32

nice you're looking for counter examples ;p
http://www.wolframalpha.com/input/?i=%5Cint_0%5E1+%28xsin%28pi+x%29%29%27%27*%28xsin%28pi+x%29%29+dx++%3D+-%5Cint_0%5E1+%28%28xsin%28pi+x%29%29%27%29%5E2dx

Answer 33

That's neat haha

Answer 34

Post questions involving Vieta's Formula so I can LEARN how to use it!!!!

Answer 35

was yesterday's drill useful jhanny ?

Answer 36

....Hah. <_<

Answer 37

Haven't heard of that in a while lol