The Mean Value theorem for derivatives and integrals?

Question

Say you compute the mean value theorem for both integrals and derivitives for the same function over the same domain. will you recieve the same awnser regardless of what f(x) is?

the mean value theorem for derivitives states that if f(X) is diferentiable and continuous at every point in the intrival [a,b], there is at least one point c in (a,b)where
f'(c) =(f(b)-f(a))/(b-a)
the mean value theorem for definite integrals states that if f is continuous on [a,b], then at some point c in (a,b)
f(c) = (b-a)^-1 * aintegralb f(x)dx

so i figure that the integral of f'(x) is f(x) + c
so (b-a)^-1 * aintegralb f(x)dx +c = aintegralb (f(b)-f(a))/(b-a)

Answer 1

No, you will not get the same answer in general. However, if F(x) is an anti-derivative of f(x), then the mean value theorem for the derivative of F(x) will give the same results as the mean value theorem for the integral of f(x). You might think about why.

Answer 2

no because the mean value theorem states that you take the average of the integral. You cant use it if youre using derivatives. so therefore, any two answers won't be the same.

Answer 3

why is the value theorem mean? ; )