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


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



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


Any ideas on how to continue in this problem?

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


I did get that


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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