# Mean Value Problem

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So I've been trying to prep for an upcoming math competition by going through some problems. This one has me a little stuck,

"Suppose $f$ is a differentiable function function on [0, 2] then there exists a point $c\in$ such that:

$f''©=f(0)-2f(1)+f(2).$"

I am not sure this statement is true under these conditions, and think twice differentiable is probably required. Assuming that $f$ is twice differentiable I have tried applying the mean value theorem, and have been able to show that there exists $a,b\in [0,2]$ and $c\in [f'(a), f'(b)]$ such that:

$f(0)-2f(1)+f(2)=f'(a)+f'(b)=f''©.$

However, I am not seeing that that $f'(a), f'(b)\in [0,2].$

Any ideas on how to continue in this problem?

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a would be between 0 and 1, while b would be between 1 and 2 (your intermediate expression should be f'(b)-f'(a)), and c between a and b, so definitely between 0 and 2.

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Sorry I was sloppy when I stated what I had done towards a solution:

I did get that

$f(0)-2f(1)+f(2)=-f'(a)+f'(b)=f''©(b-a).$

where c is in [0, 2]. But the there is still the (b-a) term that needs to be taken care of somehow, and that is where I was stuck. Sorry for the poor description in the OP.

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I haven't tried to work it out, but I believe an approach would be as follows: The mean value theorem is essentially a corollary of Rolle's theorem. The is a generalization of Rolle's theorem for higher derivatives, which could be used to answer your question.

http://en.wikipedia.org/wiki/Rolle%27s_theorem

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