My questions are very basic. I will not use fancy math language, because I am not familiar with it.
I have some function $f$ and want to calculate its monotonicity and local extreme values by using the derivative.
- First I find the domain $D_{f}$.
- Then I find the derivative $f'$.
- Then I find the domain $D_{f'}$ of the derivative $f'$.
- Then I solve the $f' = 0$ equation. Let's say the solution is only one: $x_{0}=15.$ (that is my local minimum or local maximum).
- And finally I can find the monotonicity/extreme value(s).
I have few questions related to the domain of function $D_{f}$ and the domain of the derivative $D_{f'}$.
- What if the solution $x_{0} = 15$ is in the domain $D_{f}$ but is not in the domain $D_{f'}$?
- What if the solution $x_{0} = 15$ is in the domain $D_{f'}$ but is not in the domain $D_{f}$?
- What if the solution $x_{0} = 15$ is in neither of the domains? (I'm pretty sure we ignore the $x_{0}$ solution then)
And, if that's not too much to ask:
- The domain of $D_{f} = (0, 20)$ and the domain of $D_{f'} = (5, 10)$. When finding the extreme values/monotonicity, should I take into account only the mutual part of both domains?
Thanks for help!