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Math Help - real analysis

  1. #1
    Newbie
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    real analysis

    1a) suppose that f is continuous on [a b] and differentiable on (a b) with f(a) = f(b) = 0. Show that for every k in R, there exists a c in (a b) with f '(c) = kf(c). (Hintconsider the function g(x) = exp^(-kx)f(x) for x in [a b]

    b) sho that f(x) = sinx is unformly continuos on R. (Hint: if |x-y|< delta, use the Mean Value Theorem on [x y].)
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  2. #2
    Math Engineering Student
    Krizalid's Avatar
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    Santiago, Chile
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    your problems are already solved, wonder why you didn't attach any attempt of solution.

    given g(x)=e^{-kx}f(x), since g(a)=g(b)=0 then exists c\in(a,b) so that g'(c)=-ke^{-kc}f(c)+e^{-kc}f'(c)=0\implies f'(c)=kf(c).

    the second one has been proved many times here, i suggest to you to use the seach engine.
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