[CALCULUS] f”(c)=g”(c)?

Question by fgdeqtg: [CALCULUS] f”(c)=g”(c)?
Hello!

Can someone please help me? I’ve struggle whole day with this question without any clue of how to answer it.

Suppose there are f and g, where f’ and g’ are continuous within [a,b] and f” and g” exist within (a,b). Suppose then f'(a)=g'(a) and f'(b)=g'(b)
AND THE QUESTION IS, how can I show that there is c within (a,b) so f”(c) = g”(c)???

I guess I need to use Rolle’s theorem and/or Middle Value Theorem in order to answer it, but can someone please show me how to do it??

Thank you!

Best answer:

Answer by Jamuke
Let H(x) be the function H(x) = f ‘(x) – g'(x) for all x in [a. b]. Then H is differentiable in (a, b), H ‘(x) = f “(x) – g”(x) in (a, b), and by the Mean Value Theorem, there is a number c is (a, b) such that:

H ‘(c) = (H(b) – H(a)) / (b – a)
= [(f ‘(b) – g'(b)) – (f ‘(a) – g'(a))] / (b – a)
= 0 / (b – a)
= 0.

So, H'(c) = 0 –> f “(c) – g “(c) = 0 –> f “(c) = g “(c), which was to be shown.

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