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Math Help - Tricky Advanced Calc Problem

  1. #1
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    Tricky Advanced Calc Problem

    Suppose that f is continuous and differentiable on [a,b], f(a) = f(b) = 0. Show that for every real number \lambda there is a real number c \in [a,b] such that f'(c) = \lambdaf(c).

    Tried to use MVT (and Rolle's) to no avail. Any suggestions?
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  2. #2
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    Quote Originally Posted by joeyjoejoe View Post
    Suppose that f is continuous and differentiable on [a,b], f(a) = f(b) = 0. Show that for every real number \lambda there is a real number c \in [a,b] such that f'(c) = \lambdaf(c).

    Tried to use MVT (and Rolle's) to no avail. Any suggestions?
    Since F is continuous on [a, b] it is integrable there. Let F(x)= \int_a^x f(t)dt and apply Rolle's theorem to f(x)- \lambda F(x).
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  3. #3
    Senior Member pankaj's Avatar
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    Let g(x)=e^{-\lambda x}f(x)

    g(a)=g(b)=0.Now use Rolle's Theorem
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