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Math Help - Derivative question

  1. #1
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    Derivative question

    Let f be a function from R to R, and f ' (x) > f (x) for all x. Suppose f(x_0) =0. Show that f (x) > 0 for all x > x_0.

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  2. #2
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    Re: Derivative question

    Quote Originally Posted by grandunification View Post
    Let f be a function from R to R, and f ' (x) > f (x) for all x. Suppose f(x_0) =0. Show that f (x) > 0 for all x > x_0.
    Since f'(x_0) > f(x_0) = 0, it follows that f is increasing at x_0. Therefore there is some interval to the right of x_0 in which f is positive. Suppose by way of getting a contradiction that f is not positive for all x > x_0. Then (by continuity, intermediate value theorem and suchlike properties) there must be a smallest point x_1>x_0 at which f(x) = 0. Thus f(x)>0 whenever x_0<x<x_1. But since f(x_0) = f(x_1) = 0 it follows from Rolle's theorem that there is a point x_2 with x_0<x_2<x_1 such that f'(x_2)=0. Then f'(x_2)<f(x_2), contradicting the fact that f'(x)>f(x) for all x>x_0.
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