Extreme values and monotonicity - understanding the relation between the the domain?












0












$begingroup$


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.




  1. First I find the domain $D_{f}$.

  2. Then I find the derivative $f'$.

  3. Then I find the domain $D_{f'}$ of the derivative $f'$.

  4. 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).

  5. 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 $D_{f'}$.




  1. What if the solution $x_{0} = 15$ is in the domain $D_{f}$ but is not in the domain $D_{f'}$?

  2. What if the solution $x_{0} = 15$ is in the domain $D_{f'}$ but is not in the domain $D_{f}$?

  3. 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:




  1. 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!






real-analysis derivatives







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$endgroup$












  • $begingroup$
    Why do you feel you need to find the domain of the derivative?
    $endgroup$
    – David G. Stork
    Jan 1 at 22:45












  • $begingroup$
    How could you solve $f'=0$ to get $x_0$ but not have $x_0 in D_{f'}$?
    $endgroup$
    – copper.hat
    Jan 1 at 22:47










  • $begingroup$
    @DavidG.Stork not sure, university teacher demands it
    $endgroup$
    – weno
    Jan 1 at 22:50










  • $begingroup$
    @copper.hat so I only have to check if $x_{0}$ belongs to $D_{f}$?
    $endgroup$
    – weno
    Jan 1 at 22:50






  • 1




    $begingroup$
    @weno: I am not really sure what you are asking. You must have $D_{f'} subset D_f$, otherwise it makes no sense. The derivative is defined in terms of the function's values. In general, unless the domain is 'nice' (the real line, an interval, etc) it is hard to give a nice characterisation.
    $endgroup$
    – copper.hat
    Jan 1 at 22:53


















0












$begingroup$


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.




  1. First I find the domain $D_{f}$.

  2. Then I find the derivative $f'$.

  3. Then I find the domain $D_{f'}$ of the derivative $f'$.

  4. 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).

  5. 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 $D_{f'}$.




  1. What if the solution $x_{0} = 15$ is in the domain $D_{f}$ but is not in the domain $D_{f'}$?

  2. What if the solution $x_{0} = 15$ is in the domain $D_{f'}$ but is not in the domain $D_{f}$?

  3. 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:




  1. 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!






real-analysis derivatives







share|cite|improve this question











$endgroup$












  • $begingroup$
    Why do you feel you need to find the domain of the derivative?
    $endgroup$
    – David G. Stork
    Jan 1 at 22:45












  • $begingroup$
    How could you solve $f'=0$ to get $x_0$ but not have $x_0 in D_{f'}$?
    $endgroup$
    – copper.hat
    Jan 1 at 22:47










  • $begingroup$
    @DavidG.Stork not sure, university teacher demands it
    $endgroup$
    – weno
    Jan 1 at 22:50










  • $begingroup$
    @copper.hat so I only have to check if $x_{0}$ belongs to $D_{f}$?
    $endgroup$
    – weno
    Jan 1 at 22:50






  • 1




    $begingroup$
    @weno: I am not really sure what you are asking. You must have $D_{f'} subset D_f$, otherwise it makes no sense. The derivative is defined in terms of the function's values. In general, unless the domain is 'nice' (the real line, an interval, etc) it is hard to give a nice characterisation.
    $endgroup$
    – copper.hat
    Jan 1 at 22:53
















0












0








0





$begingroup$


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.




  1. First I find the domain $D_{f}$.

  2. Then I find the derivative $f'$.

  3. Then I find the domain $D_{f'}$ of the derivative $f'$.

  4. 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).

  5. 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 $D_{f'}$.




  1. What if the solution $x_{0} = 15$ is in the domain $D_{f}$ but is not in the domain $D_{f'}$?

  2. What if the solution $x_{0} = 15$ is in the domain $D_{f'}$ but is not in the domain $D_{f}$?

  3. 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:




  1. 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!






real-analysis derivatives







share|cite|improve this question











$endgroup$




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.




  1. First I find the domain $D_{f}$.

  2. Then I find the derivative $f'$.

  3. Then I find the domain $D_{f'}$ of the derivative $f'$.

  4. 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).

  5. 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 $D_{f'}$.




  1. What if the solution $x_{0} = 15$ is in the domain $D_{f}$ but is not in the domain $D_{f'}$?

  2. What if the solution $x_{0} = 15$ is in the domain $D_{f'}$ but is not in the domain $D_{f}$?

  3. 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:




  1. 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!






real-analysis derivatives




real-analysis derivatives






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 1 at 22:38









Bernard

119k740113




119k740113










asked Jan 1 at 22:31









wenoweno

1349




1349












  • $begingroup$
    Why do you feel you need to find the domain of the derivative?
    $endgroup$
    – David G. Stork
    Jan 1 at 22:45












  • $begingroup$
    How could you solve $f'=0$ to get $x_0$ but not have $x_0 in D_{f'}$?
    $endgroup$
    – copper.hat
    Jan 1 at 22:47










  • $begingroup$
    @DavidG.Stork not sure, university teacher demands it
    $endgroup$
    – weno
    Jan 1 at 22:50










  • $begingroup$
    @copper.hat so I only have to check if $x_{0}$ belongs to $D_{f}$?
    $endgroup$
    – weno
    Jan 1 at 22:50






  • 1




    $begingroup$
    @weno: I am not really sure what you are asking. You must have $D_{f'} subset D_f$, otherwise it makes no sense. The derivative is defined in terms of the function's values. In general, unless the domain is 'nice' (the real line, an interval, etc) it is hard to give a nice characterisation.
    $endgroup$
    – copper.hat
    Jan 1 at 22:53




















  • $begingroup$
    Why do you feel you need to find the domain of the derivative?
    $endgroup$
    – David G. Stork
    Jan 1 at 22:45












  • $begingroup$
    How could you solve $f'=0$ to get $x_0$ but not have $x_0 in D_{f'}$?
    $endgroup$
    – copper.hat
    Jan 1 at 22:47










  • $begingroup$
    @DavidG.Stork not sure, university teacher demands it
    $endgroup$
    – weno
    Jan 1 at 22:50










  • $begingroup$
    @copper.hat so I only have to check if $x_{0}$ belongs to $D_{f}$?
    $endgroup$
    – weno
    Jan 1 at 22:50






  • 1




    $begingroup$
    @weno: I am not really sure what you are asking. You must have $D_{f'} subset D_f$, otherwise it makes no sense. The derivative is defined in terms of the function's values. In general, unless the domain is 'nice' (the real line, an interval, etc) it is hard to give a nice characterisation.
    $endgroup$
    – copper.hat
    Jan 1 at 22:53


















$begingroup$
Why do you feel you need to find the domain of the derivative?
$endgroup$
– David G. Stork
Jan 1 at 22:45






$begingroup$
Why do you feel you need to find the domain of the derivative?
$endgroup$
– David G. Stork
Jan 1 at 22:45














$begingroup$
How could you solve $f'=0$ to get $x_0$ but not have $x_0 in D_{f'}$?
$endgroup$
– copper.hat
Jan 1 at 22:47




$begingroup$
How could you solve $f'=0$ to get $x_0$ but not have $x_0 in D_{f'}$?
$endgroup$
– copper.hat
Jan 1 at 22:47












$begingroup$
@DavidG.Stork not sure, university teacher demands it
$endgroup$
– weno
Jan 1 at 22:50




$begingroup$
@DavidG.Stork not sure, university teacher demands it
$endgroup$
– weno
Jan 1 at 22:50












$begingroup$
@copper.hat so I only have to check if $x_{0}$ belongs to $D_{f}$?
$endgroup$
– weno
Jan 1 at 22:50




$begingroup$
@copper.hat so I only have to check if $x_{0}$ belongs to $D_{f}$?
$endgroup$
– weno
Jan 1 at 22:50




1




1




$begingroup$
@weno: I am not really sure what you are asking. You must have $D_{f'} subset D_f$, otherwise it makes no sense. The derivative is defined in terms of the function's values. In general, unless the domain is 'nice' (the real line, an interval, etc) it is hard to give a nice characterisation.
$endgroup$
– copper.hat
Jan 1 at 22:53






$begingroup$
@weno: I am not really sure what you are asking. You must have $D_{f'} subset D_f$, otherwise it makes no sense. The derivative is defined in terms of the function's values. In general, unless the domain is 'nice' (the real line, an interval, etc) it is hard to give a nice characterisation.
$endgroup$
– copper.hat
Jan 1 at 22:53












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