Solving Integration Question: f' Continuous on [a,b]

[trap]

I'm having major trouble with this question, can anyone assist me on this?

Let f be a function such that f' is continuous on [a,b]. Show that

\int_a^{b} f(t)f’(t) dt = 1/2 [f''(b) - f''(a)]

Hint: Calculate the derivative of F(x) = f''(x).

 

[Zurtex]

Don't double post please

 

[trap]

sorry, i thought a different forum would make a difference, since no one has an answer to my question yet.

 

[Zurtex]

sorry, i thought a different forum would make a difference, since no one has an answer to my question yet.

Have you not read your other thread? People have been quite helpful.

 

[shmoe]

Hi, are you sure those are second derivatives on the right hand side and not squares? Something like 1/2(f(b)^2-f(a)^2) instead?

 

[trap]

i'm not sure if they are squares becoz the question reads f^2(b) - f^2(a)..so i thought they were second derivative..

 

[Zurtex]

i'm not sure if they are squares becoz the question reads f^2(b) - f^2(a)..so i thought they were second derivative..

That is squares and I am sure about that because it is the answer. When your talking about the nth derivative you either use roman numerals or put the number in brackets.

 

[DeadWolfe]

Couldn't one simply integrate by parts to get the answer?

 

[digink]

Hi, are you sure those are second derivatives on the right hand side and not squares? Something like 1/2(f(b)^2-f(a)^2) instead?

Could you please explain how you come to that answer, I am having trouble seeing it.

Thanks.

 

[shmoe]

Could you please explain how you come to that answer, I am having trouble seeing it.

Thanks.

Just use the hint applied to F(x)=(f(x))^2. Find the derivative of F(x) using the chain rule...


DeadWolfe-yes integration by parts will work fine.