How to tell if a function's derivative is always positive?

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Discussion Overview

The discussion revolves around determining whether the derivative of a given function, specifically \(\frac{dP}{dt}=P(a-bP)\), is always positive or negative between its critical points \(P=0\) and \(P=a/b\). Participants explore the reasoning behind testing points within the interval defined by these critical points and the implications of the function's behavior.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants note that the critical points of the function are \(P=0\) and \(P=a/b\) and suggest testing an arbitrary point, such as \(P=\frac{a}{2b}\), to determine the sign of the derivative.
  • One participant describes the function \(P(a-bP)\) as a parabola, indicating that its vertex lies between the critical points and that the sign of the derivative must be consistent (either always positive or always negative) between these points.
  • Another participant questions the reasoning behind the assertion that the derivative cannot change sign between the critical points, suggesting that the continuity and definition of the function in that interval are crucial.
  • A later reply confirms that the derivative does not equal zero between the critical points, reinforcing the idea that it cannot change sign, provided the function is defined and continuous in that interval.

Areas of Agreement / Disagreement

Participants generally agree on the method of testing points to determine the sign of the derivative, but there is some uncertainty regarding the reasoning behind the behavior of the derivative between the critical points. The discussion remains unresolved regarding the clarity of this reasoning.

Contextual Notes

Participants mention the importance of the function being defined and continuous between the critical points, but do not elaborate on specific conditions or assumptions that may affect the analysis.

find_the_fun
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In class we were given an example where [math]\frac{dP}{dt}=P(a-bP)[/math]. We found the critical points to be P=0 and P=a/b. We wanted to know if the derivative is always positive or negative between the two critical points. The prof said you could pick an arbitrary point between the two, such as [math]\frac{a}{2b}[/math] and plug that into the derivative and check to see if it's greater than 0. So [math]P'(\frac{2}{2b})=\frac{a^2}{4b} > 0[/math].

I'm really confused and don't understand the reasoning, can someone fill in the gaps?
 
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find_the_fun said:
In class we were given an example where [math]\frac{dP}{dt}=P(a-bP)[/math]. We found the critical points to be P=0 and P=a/b. We wanted to know if the derivative is always positive or negative between the two critical points. The prof said you could pick an arbitrary point between the two, such as [math]\frac{a}{2b}[/math] and plug that into the derivative and check to see if it's greater than 0. So [math]P'(\frac{2}{2b})=\frac{a^2}{4b} > 0[/math].

I'm really confused and don't understand the reasoning, can someone fill in the gaps?

Hi find_the_fun,

$$P(a-bP)$$ is a parabola.
Its top is between $P=0$ and $P=a/b$, and actually at $P=\frac{a}{2b}$.
Either way, the sign of the derivative, which is equal to $P(a-bP)$, between those 2 points is either always positive, or always negative.
Testing it at some point between those boundaries will tell us which it is.

Alternatively, we can see that if $b >0$, the parabola is upside-down, meaning it is always positive between the 2 points.
 
I like Serena said:
the sign of the derivative...between those 2 points is either always positive, or always negative.

How do you know this? Is it because we know [math]\frac{dP}{dt}[/math] is not 0 between the two points, therefore it can't change sign?
 
Last edited:
find_the_fun said:
How do you know this? Is it because we know [math]\frac{dP}{dt}[/math] is not 0 between the two points, therefore it can't change sign?

Yes.
(With the additional information that it is defined and continuous between those 2 points.)
 

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