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## Homework Statement

Suppose f is differentiable on [a, b], f(a) = 0, and there is a real number A such that |f '(x)

≤ A |f(x)| on [a, b]. Prove f(x) = 0 for all x in [a, b].

## Homework Equations

Might need the Mean Value Theorem or a variation thereof.

## The Attempt at a Solution

Suppose instead that there is an x

_{0}in [a, b] such that |f '(x

_{0})| > 0.

Then we have

0 < |f(x

_{0})| = |f(x

_{0}) - f(a)| ≤ (x

_{0}- a) |f '(x)| ≤ A(x

_{0}- a)|f(x)|

for some x in (a, x

_{0}).

Am I close? Give me the vaguest of hints, as I am really not supposed to solicit help on this assignment.

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