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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Since , it follows that f is increasing at Therefore there is some interval to the right of in which f is positive. Suppose by way of getting a contradiction that f is not positive for all Then (by continuity, intermediate value theorem and suchlike properties) there must be a smallest point at which Thus whenever But since it follows from Rolle's theorem that there is a point with such that Then , contradicting the fact that for all