Exercise with lagrange and derivatives

[Felafel]

Homework Statement

Being a>0 and f:[a,b]--->R continuos and differentiable in (a,b), show that there exists a t ##\in## (a,b) such that:
## \frac{bf(a)-af(b)}{b-a}=f(t)-tf'(f)##

The Attempt at a Solution

For lagrange's theorem, we have:
## \frac{f(a)-f(b)}{b-a}= -f'(t) ##

thought i could find f(t) from the equation ##f(x) = f(t) + f'(t)(x-t)+R_1(x-t)## ignoring R.
but then a "x" would appear and I don't know how to deal with it.

 

[mfb]

To simplify the equation, consider g:[a,b]->R with
$$g(x)=f(x)-f(a)-(x-a)\frac{f(b)-f(a)}{b-a}$$
This is chosen to get g(a)=g(b)=0 and ##g'(x)=f'(x)-\frac{f(b)-f(a)}{b-a}##

Is there some t ∈ (a,b) such that ##0=g(t)-tg'(t)##?
Working backwards, you should be able to get the original equation.

 

[Felafel]

sounds good, but I'm explicitly asked to use Lagrange's theorem

 

[mfb]

I think you can use that theorem to solve the reduced problem ;).

 

[Felafel]

another hint? i don't get it, i always end up in a loop :(