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

[find_the_fun]

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?

 

[I like Serena]

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.

 

[find_the_fun]

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?

 

[I like Serena]

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.)