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Where does it follow from that the derivative of a differentiable function on a closed interval is bounded above/below? Specifically, let $f: [a, b] \to \mathbb{R}$ by $f(x) = cx^n, n \in \mathbb{N}$. By what property is $f'(x)$ bounded on $(a,b)$ so that is has a supremum?

Epsilon Away
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    You are lucky that $f'$ is continuous on this interval, so that $f'$ is continuous on a closed and bounded interval, hence is bounded. But in general, $f'$ need not be bounded. See here , for example. – Sarvesh Ravichandran Iyer Jan 28 '21 at 06:39
  • @TeresaLisbon I see. So we could just make the extrapolation from the interval of $(a, b)$ to $[a, b]$, since $f$ is continuous on the latter, and state that as a continuous on a closed and bounded interval, $f'$ attains both its maximum and minimum value? – Epsilon Away Jan 28 '21 at 06:51
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    Yes that is what I meant – Sarvesh Ravichandran Iyer Jan 28 '21 at 06:54

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