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Topic: Having a sleep problem by mean value theorem
Replies: 2   Last Post: Nov 29, 2013 6:45 AM

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Torsten Hennig

Posts: 2,419
Registered: 12/6/04
Re: Having a sleep problem by mean value theorem
Posted: Nov 28, 2013 4:33 AM
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> Dear professors,
> I have some question on mean value theorem. This is
> not a homework, but rather my curiosity. If answered,
> I would be very happy.
> Suppose that f1(t), f2(t) are some continuous
> real functions of the real variable t on an interval
> [a, b] and that both are differentiable on (a, b).
> Put F(t) = f1(t) + f2(t). Then, by the mean value
> theorem,
> there is a number c in (a, b) for which
> F'(c) = [f1(b) - f1(a)]/(b - a) + [f2(b) - f2(a)]/(b
> - a).
> ..(1)
> My question is, does this equation (1) imply
> f1'(c) = [f1(b) - f1(a)]/(b - a), ...(2) and
> f2'(c) = [f2(b) - f2(a)]/(b - a), ...(3)
> respectively?
> First, I thought this is a kind of trivial question,
> and the answer to this question should be yes.
> But the more I think about it, I can't
> find any good explanation for the reason the answer
> should be yes. Now I am a bit doubtful of the
> question.
> Maybe you could help me by giving a counterexample
> to this question.
> Please help me on this question for my better sleep.
> P.

Try f1(x)=x^2 and f2(x)=x^3, and you'll see that the answer to your question is "no".

Best wishes

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