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 October 1st, 2013, 12:39 PM #1 Newbie   Joined: Oct 2013 Posts: 5 Thanks: 0 Applying Rolle's Theorem Prove that if a function$f(x)$ is continuous on a segment [a,b] and $\int_a^b f(x)dx= 0$, then there exists a point $c \in (a,b)$ such that $\int_a^c f(t)dt= f(c)$. Hint: Apply Rolle's Theorem to the function $g(x)= e^{-x}\int_a^x f(t)dt$ where $(a \leq x \leq b)$.
 October 1st, 2013, 01:49 PM #2 Senior Member   Joined: Jun 2013 From: London, England Posts: 1,312 Thanks: 115 Re: Applying Rolle's Theorem g(a) = g(b) = 0, hence by Rolle's theorem, there exists c, such that g'(c) = 0. $g'(x) = -e^{-x}\int_a^x f(t)dt \ + \ e^{-x}f(x)$ $0 \= \ g#39;(c) = -e^{-c}\int_a^c f(t)dt \ + \ e^{-c}f(c)$ Hence: $\int_a^c f(t)dt \= \ f(c)$

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