I'm working on the following problem:

Let $h$ be a differentiable function defined on the interval $[0, 3]$, and assume that $h(0) = 1, h(1) = 2$, and $h(3) = 2$. Argue that $h'(x) = 1/4$ at some point in the domain.

I've tried using mean value property and the generalized mean value by guessing some other function. I can obtain that $h'$ will attain the values $0$ and $1/3$. I was going to argue by IVT that $1/4$ will be attained, but I don't think I can assume the derivative is continuous -- so I'm not sure how to proceed.


You should read up on Darboux's theorem, which says that derivatives have the intermediate value property, even if they are NOT continuous. (I love this theorem!) en.wikipedia.org/wiki/Darboux%27s_theorem_(analysis)

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  • $\begingroup$ you may wish to fix that URL $\endgroup$ – zhw. Apr 13 '18 at 1:27

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