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Let $f:\Bbb{R}\rightarrow \Bbb{R}$, $f$ continuous and differentiable. When is it true that for $a<b$ that: $$f'(a)\leq \frac{f(b)-f(a)}{b-a}\leq f'(b)$$

So the motivation for this was in comparing average velocity, which is given by the slope of secant line, to the instantaneous velocities at the endpoints, given by derivatives, in a 1-d motion problem.

So naturally, if $f'$ is increasing on $[a,b]$ we can apply the mean value theorem to say that there is an $x \in (a,b)$ such that $f(x)=\cfrac{f(b)-f(a)}{b-a},$ and since $a<x<b$, and $f'$ is increasing we're done.

Are these conditions too strict?

I suppose the more general question is when should one expect either of the instantaneous quantities, or the secant slope to be larger?

Just as an example for a function which is not increasing:

$f(x)=2x^3-3x^2$, whereby $f'(0)=f'(1)=0$ whilst the secant line slope is $-1$. In this case $f'$ is decreasing in the interval.

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  • $\begingroup$ We will need some sort of "global" condition such as your $f'(x)$ increasing, for $f'(a)$ and $f'(b)$ can be modified arbitrarily without affecting $f(b)-f(a)$ significantly. $\endgroup$ – André Nicolas Oct 5 '15 at 21:09
  • $\begingroup$ Without any conditions, all three values can be picked at random, and a function $f$ exists which has them. Both of your examples are monotone, so there is still a relationship that can be established, but if you don't require a monotone function, it can start off heading in any direction, then turn around if needed to head for the other point. $\endgroup$ – Paul Sinclair Oct 5 '15 at 22:28

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