Applying Rolle's Theorem

[bobcantor1983]

Prove that if a function \(\displaystyle f(x)\) is continuous on a segment [a,b] and \(\displaystyle \int_a^b f(x)dx = 0\), then there exists a point \(\displaystyle c \in (a,b)\) such that \(\displaystyle \int_a^c f(t)dt = f(c)\).


Hint: Apply Rolle's Theorem to the function \(\displaystyle g(x) = e^{-x}\int_a^x f(t)dt\) where \(\displaystyle (a \leq x \leq b)\).

Any guidance on how to proceed here would be very helpful.

 

[HallsofIvy]

Have you even tried using the hint? \(\displaystyle g(a)= e^{-a}\int_a^a f(t)dt= 0\) and \(\displaystyle g(b)= e^{-b}\int_a^b f(t)dt= 0\). Certainly, by the "fundamental theorem of Calculus", g is differentiable so the "Mean value theorem" says that, for some c between a and b, the derivative of g is 0. What is the derivative of \(\displaystyle e^{-x}\int_a^x f(t)dt\)? (Use the product rule and the fundamental theorem of Calculus).