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Let $f:[0,\frac{1}{2}] \rightarrow\mathbb{R}$ be a differentiable function such that $|f^\ {'}(x)|\leq |f(x)|$ and $f(0)=0.$ Prove: $f(x)=0, \forall x\in [0,\frac{1}{2}]$.

So I tried to work by the definition of the derivative at $0$ and somehow squeeze it, but got stuck.

Any help appreciated.

Martin Sleziak
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Itay4
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    Also: http://math.stackexchange.com/questions/1052785/consequence-of-mean-value-theorem, http://math.stackexchange.com/questions/509876/prove-fx-0-for-all-x-in-0-infty-when-fx-leq-fx, http://math.stackexchange.com/questions/816021/prove-that-fx-equiv0-on-left0-1-right – Martin R Mar 02 '17 at 09:48
  • Related: http://math.stackexchange.com/questions/1383413/condition-to-guarantee-f-0-on-a-b. – Martin R Mar 02 '17 at 10:44

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