Is the derivative of a monotonically increasing function positve?

Question

I know that if the derivative of a function is positive on $\left(a,b\right)$ then that function is increasing on $\left(a,b\right)$ . However not sure if the opposite always holds. Here is my approach:

Let $f\left(x\right)$ monotone increasing on $\left(a,b\right)$ Then for every $a_1$ and $b_1$ of $\left(a,b\right)$ such that if $a_1<b_1$, it holds that $f\left(a_1\right)<f\left(b_1\right)$,

So we have $$ \frac{f\left(b_1\:\right)-f\left(a_1\right)}{b_1-a_1}>0\:$$

From Langranges' theorem we have that there exists a $c$ in $\left(a_1,b_1\right)$ such that

$$ f'\left(c\right)\:=\:\frac{f\left(b_1\:\right)-f\left(a_1\right)}{b_1-a_1}>0\: $$ This means that for every subsegment of $\left(a,b\right)$, we can find a $c$, such that it's derivative at $c$ is positive. Meaning that the derivative of $f(x)$ on $\left(a,b\right)$ is always positive for every point in $\left(a,b\right)$.

You don't get to choose $c$, so you don't know that every value in $(a,b)$ is covered by this argument. — dxiv, Dec 09 '21 at 22:23
Looks fine assuming the function is differentiable everywhere on $(a,b)$. — CyclotomicField, Dec 09 '21 at 22:29
You want to start with some point $x\in(a,b)$ and show that $f'(x)>0$. — Andrea B., Dec 09 '21 at 22:33
Does this answer your question? $f$ strictly increasing does not imply $f' \gt 0$ — dxiv, Dec 09 '21 at 22:34
$f(x) = x^3$ is monotonically increasing, but the derivative is not positive at $x=0$. — Malcolm, Dec 09 '21 at 22:35
Thank you for your answers. Does my proof however guarantee that there will be at least one point at which the derivative will be positive? — Drilon Aliu, Dec 09 '21 at 22:38
Imagine "$x^3$" cut and glued over and under cut "$x^3$" . So can be done infinite times - sorry if idea is badly explained - can you imagine it? — zkutch, Dec 09 '21 at 23:36
@AlexParker Yes, and you can in fact guarantee infinitely many points where $f'$ is strictly positive, since you can recursively apply MVT to $,(a,c), (c,b),$. — dxiv, Dec 10 '21 at 00:53

score 2 · Answer 1 · answered Dec 09 '21 at 22:34

A function strictly increasing need not have a (strictly) positive derivative at all points. For instance $f(x)=x^3$ is monotonically (strictly) increasing on the interval $(-1,1)$, but the derivative is not always positive on that interval: Namely: $f'(0)=0$.

Is the derivative of a monotonically increasing function positve?

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