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I have to prove the following statement:

$f$ is twice differentiable on $[a, a+h]$. Prove that there exists $c \in (a, a+h)$ such that $$f(a+h)=f(a)+hf'(a)+\frac{1}{2}h^2f''(c)$$

I need to solve this problem without using Taylor's theorem, L'Hopital's rule, or any higher order of thinking. The things that I can use is the mean value theorem, Cauchy's MVT, and definition of differentiation. I tried to mimic the proof for the Lagrange form of Taylor's Theorem that uses Cauchy's MVT, given in this page.(Wikipedia_Page) Also I tried to apply the MVT twice, but it did not give me the exact form, instead the term $h(b-a)f''(c)$. I tried to view the RHS as a quadratic polynomial about $h$, and tried to use IVT for the cases where $c=a$ and $c=a+h$. All did not give me a satisfactory proof.

How can I get the required result? Are any of my approaches able to lead to the proof?

Joshua Woo
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  • Check this: https://math.stackexchange.com/q/481661/42969 – Martin R Jul 27 '21 at 04:18
  • Or this: https://en.wikipedia.org/wiki/Taylor%27s_theorem#Derivation_for_the_mean_value_forms_of_the_remainder – Martin R Jul 27 '21 at 04:20
  • @MartinR Thanks for the link. Actually the second link was the same page in my post. I'll give it another try. I am now pretty sure that the proof of Taylor's Theorem at $k=1$ is the case I'm looking for. – Joshua Woo Jul 27 '21 at 04:24
  • The Wikipedia page shows a proof using Cauchy's mean value theorem. – Martin R Jul 27 '21 at 04:44
  • @MartinR Yes indeed. Thanks to your help, I solved the problem. Thanks a lot sir. – Joshua Woo Jul 27 '21 at 04:46

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