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# Divide by derivative

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Hi friends,

I understand that when we want to get x values for the following:

$$f(x).f'(x)>0$$

we look at the sections where both graphs are either positive or negative

$$f(x).f'(x)<0$$

we look at all sections where one graph is positive or negative and the other, negative or positive, respectively.

$${f'(x) \over{f(x)}}<0...?$$

Or greater than zero?

This question aplies to $$f(x)=3x^3-21x^2+45x-27$$

Thank you kindly..

Oct 15, 2020
edited by juriemagic  Oct 15, 2020

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Same story, one of the two has to be negative and the other has to be positive .....and neither can be zero.

For GREATER than zero both have to be positive  ....or both have to be negative.

DO you need more than that?

Oct 15, 2020
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ElectricPavlov,

so then...there will be no difference between the answers of

$$f(x).f'(x)<0$$

and

$${f(x) \over{f'(x)}}<0$$

because in both instances the criteria is the same...one equation must be negative or positive, while the other is positive or negative, respectively?

juriemagic  Oct 15, 2020
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Numerically, there will be different answers.

Example:

-12 * 3 = -36

-12/3 = -4

But I think the domain expressions will be the same.....

ElectricPavlov  Oct 15, 2020
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Thank you kindly...just was not sure about the domain expressions..do appreciate.

juriemagic  Oct 15, 2020
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The 'domain expressions' are the areas ..or values...of   'x'   that satisfy your equation..

like from x = 3 to infinity     (  3 < x < + inf)

or   from   x = 1.5 to 3     which would be  ( 1.5 < x < 3)

ElectricPavlov  Oct 15, 2020
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ElectricPavlov,

yes, I understand all that, I was just having trouble with understanding the difference between multiplying two functions to yield a specific outcome, and deviding those functions for a similar outcome....

juriemagic  Oct 15, 2020
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Look at the graph and see if you can decide from that:

Here are graphs of the individual functions:

Oct 15, 2020
edited by Alan  Oct 15, 2020
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Alan,

I'm sorry, no..I do not see what I should see there..

juriemagic  Oct 15, 2020
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In the second graph you can see where the individual functions are + or -   ....

or where they are both -    or   both +

If you look closely wou will see   +  and -   up until approx x = '1' then a little bit of both + until x ~ 1.5  then   + - again up until 3

then both +   above 3

this is graphed in the upper picture  you can see where the division of the two is negative   then a small bit of positive...then negative up to x=3   then positivr above x = 3

ElectricPavlov  Oct 15, 2020
edited by ElectricPavlov  Oct 15, 2020
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Thank you ElectricPavlov...I will study that a bit closer....do appreciate..

juriemagic  Oct 15, 2020