Average Value of a function

1. Jul 24, 2011

dohsan

1. The problem statement, all variables and given/known data

If fave [a,b] denotes the average value of f on the interval [a,b] and a<c<b, show that
fave[a,b] = (c-a)/(b-a) fave[a,c] + (b-c)(b-a) fave[c,b]

2. Relevant equations

All i know is the mean value theorem for integrals is f(c) = fave = 1/(b-a) integral(f(x),x,b,a)

3. The attempt at a solution

Tried using the theorem, but had no idea how to get to that point.

Thanks!

2. Jul 24, 2011

antny

Hi,

Why don't you apply the definition of the fave to fave[a,c] and fave[c,b]?

Once you do this, I think you'll see that your expression simplifies quite nicely.

3. Jul 24, 2011

dohsan

Hello, I tried it and i believe this is how it goes...

fave[a,c] = 1/(c-a) [f(c)-f(a)]
fave[c,b] = 1/(b-c) [f(b)-f(c)]

then i add it together or what? Kind of confused on what to do because this gives something weird...

I do know that u can split up the bounds from [a,c] and [c,b] to get [a,b].. does that correlate with what this gotta do?

4. Jul 24, 2011

antny

Perhaps I don't understand your notation, but shouldn't [f(c)-f(a)] be Int[f, a, c]?

Try plugging in those expressions into the right-side of the equation that you're trying to prove.