Positive derivative and strictly increasing

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Discussion Overview

The discussion revolves around the implications of a positive derivative for a function, specifically whether a function is strictly increasing on an interval when its derivative is positive. Participants explore the relationship between the derivative and the behavior of the function, as well as the conditions under which this relationship holds.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant asserts that if a function f(t) has a positive derivative f'(t) > 0 on an interval, then the function is strictly increasing on that interval, meaning s < t implies f(s) < f(t).
  • Another participant suggests that this can be proven using the mean value theorem by assuming points s and t with s < t and f(s) ≥ f(t), leading to a contradiction.
  • A third participant provides a mathematical expression relating to the derivative, indicating that if x2 > x1 implies f(x2) > f(x1), then the derivative f'(x) must be positive.
  • One participant expresses gratitude for the discussion, indicating they were unsure about the mean value theorem but appreciated the clarification.
  • A different participant questions whether a function with f'(a) > 0 necessarily has an interval of positive length centered at c where the function is strictly increasing, proposing that the answer might be no and suggesting the existence of oscillating functions as a potential counterexample.

Areas of Agreement / Disagreement

Participants generally agree on the implications of a positive derivative for strict monotonicity in specific intervals, but there is disagreement regarding the existence of intervals of positive length around points where the derivative is positive. The discussion remains unresolved regarding the counterexample proposed.

Contextual Notes

The discussion includes assumptions about the behavior of functions and their derivatives, but does not resolve the conditions under which these assumptions hold, particularly in relation to oscillating functions.

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Isn't it true that if a function [itex]f(t)[/itex] satisfies [itex]f'(t) > 0[/itex] on some interval, then [itex]f[/itex] is STRICTLY increasing on that interval; i.e., that [itex]s < t[/itex] implies that [itex]f(s) < f(t)[/itex]?
 
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You can prove this using the mean value theorem, by assuming there are points s and t with [tex]s<t[/tex] and [tex]f(s)\geq f(t)[/tex] and developing a contradiction
 
Yes.
In fact:
if x2>x1 implies that
f(x2)>f(x1)the derivative is:
[f(x2)-f(x1)]/(x2-x1)
(as x2 approach x1)

The numerator is positive and also the denominator (by hypothesis ) implies that:
f'(x)>0
 
Thanks, guys. I was pretty sure this was a MVT exercise, but I was so tired last night I couldn't force myself to get out of bed and look for paper and pencil to run through it :) I'm amazed I was actually able to phrase the question correctly.
 
Related to the topic at hand, if, on the other hand, f'(a) > 0, is there necessarily an interval of positive length that is centered at c, on which f is strictly increasing? I've though about this for a while now, and it seems to me the answer is no, but I just can't come up with a counterexample. I can imagine an ever oscillating function that never reaches f(a) when x > a, but since it constantly oscillates, you also can't pinpoint the endpoint of an interval, on which f would be strictly increasing.

Any thoughts on this, is this a good counterexample? And does anyone perhaps have a specific function in mind, one that isn't just vaguely described such as above in my previous paragraph?
 
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