Question

proof that mean value theorem for definite integral.

Answer 1

Original mean value theorem: \[
f(b)-f(a)=f'(x^*)(b-a)
\]Using the fundamental theorem of calculus: \[
\int_a^bf'(x)\;dx=f'(x^*)(b-a)
\]If we let \(g(x)=f'(x)\) then: \[
\int_a^b g(x)\;dx=g(x^*)(b-a)
\]You could also write this as: \[
g(x^*)=\frac{1}{b-a}\int_a^b g(x)\;dx

\]

Answer 2

So basically you have: \[
f'(x^*)=\frac{f(b)-f(a)}{b-a}
\]And \[
g(x^*)=\frac 1{b-a}\int_a^bg(x)\;dx
\]

Answer 3

@wio thank you very much