> 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".