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Mean Value Problem Competition Practice Rate Topic: -----

#1 16 January 2012 - 07:48 PM DJBruce 


Molecule
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?
"To give anything less than your best is to sacrifice the Gift."

"A lot of people run a race to see who is fastest. I run to see who has the most guts, who can punish himself into exhausting pace, and then at the end, punish himself even more."
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#2 16 January 2012 - 10:26 PM mathematic 


Atom
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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#3 17 January 2012 - 12:26 AM DJBruce 


Molecule
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.
"To give anything less than your best is to sacrifice the Gift."

"A lot of people run a race to see who is fastest. I run to see who has the most guts, who can punish himself into exhausting pace, and then at the end, punish himself even more."
0

#4 17 January 2012 - 11:49 PM mathematic 


Atom
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....lle%27s_theorem
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