(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove that if f(x) is continuous and [tex]f(x) = \int_0^x f(x) dx[/tex], then f(x) = 0.

2. Relevant equations

3. The attempt at a solution

If [tex]f(x) = \int_0^x f(x) dx[/tex], then by integrating by the FTC we have f'(x) = f(x). Thus the only solution to this equation will have the form [tex]f(x) = ce^x[/tex] for some constant c. Now, [tex]f(x) = \int_0^x f(x) dx = f(x) - f(0) [/tex], implying that f(0 = 0. So since we know the solution to the equation will be [tex] f(x) = ce^x[/tex] then we have [tex] 0 = f(0) = ce^0 = c[/tex], implying that c = 0. Thus f(x) = 0. QED

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# Homework Help: Derivative/Integral proof

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