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Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.

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Prove or disprove: A differentiable function $f$ is always non-negative with this condition

I want to prove that a differentiable function $f: [0,\infty) \rightarrow \mathbb{R} $, which satisfies the following condition is always non-negative: Assume $f(0)=0$ and whenever $f(a)=0$, then $f' …
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