2

I am reading the mean value theorem and its corollaries.
I am referring in particular to the corollary which says that
if $f'$ is non-negative then the function is increasing.

And... one thing here got me thinking about the following corner case scenario.

So... is it possible to construct a function (the simpler the function, the better)

$f : [a, +\infty) \rightarrow \mathbb{R} $

which is differentiable in this interval and

$f'(a) \lt 0$ but $f'(x) \ge 0$ for all $x \gt a$

In this case would the corollary of the mean value theorem be applicable i.e.
would we be able to claim that the function $f$ is increasing (in this whole interval)?

In other words to be able to apply the corollary and claim that the function is increasing... does the value of the derivative matter in the boundary point of the interval? Or... what matters is only that the derivative is non-negative in all internal points of the interval?

Or... (third option) maybe the construction I am asking about is not possible at all? But... the construction should be possible, no? Simply the function $f'$ needs to be discontinuous at the point $a$, right?

peter.petrov
  • 12,568
  • 1
    This could be worth reading: https://math.stackexchange.com/a/1746219 – Minus One-Twelfth May 30 '20 at 11:18
  • 1
    @MinusOne-Twelfth Wow... now I am a bit confused... So a derivative cannot be discontinuous in the way I describe it here? Because... the construction I had in my mind involves a jump discontinuity, right? I have never heard of this theorem of Darboux, I think. I wonder why it's not mentioned in my real analysis book. Very interesting! Thanks. – peter.petrov May 30 '20 at 11:30
  • 2
    @peter.petrov In fact all derivatives satisfy the intermediate value property. – Allawonder May 30 '20 at 11:41
  • 1
    @Allawonder Then (in the context of my question) based on Darboux's theorem, is it right to say that if $f'(x) \ge 0$ for all $x \gt a$ then $f'(a)$ either does not exist or $f'(a)$ is also non-negative? I think that's correct, right? – peter.petrov May 30 '20 at 12:23
  • 1
    @peter.petrov Exactly! That would be the right conclusion. – Allawonder May 30 '20 at 12:28
  • 1
    @Allawonder OK, I think I got it. Thanks. – peter.petrov May 30 '20 at 12:30

1 Answers1

1

Mean value theorem forbids what you are trying to construct. Consider the ratio $$\frac{f(a+h) - f(a)} {h} $$ for $h>0$. By mean value theorem this equals $f'(\xi) $ for some $\xi\in(a, a+h) $ and hence is non-negative. And hence if this ratio tends to a limit then the limit must be non-negative. And this means $f'(a) \geq 0$.

Paramanand Singh
  • 87,309
  • Thanks, that is quite a simple argument so... quite nice, thanks. – peter.petrov May 31 '20 at 19:36
  • @peter.petrov: glad that you liked it. Mean value theorem can also be used to prove the intermediate value property of derivatives (Darboux theorem discussed in comments to your question). – Paramanand Singh Jun 01 '20 at 01:27