Finding increasing/decreasing intervals of an equation using critical points?

[shocklightnin]

Homework Statement

Hi I have an equation as follows:
f(x) = (2x-2.3)/(2x-5.29)^2

what i got for the derivative was:
f'(x) = (-1.38-4x)/(2x-5.29)^3

Homework Equations

f(x) = (2x-2.3)/(2x-5.29)^2
f'(x) = (-1.38-4x)/(2x-5.29)^3

The Attempt at a Solution

what i got for the critical point is -0.345, but then the question asks when the function is increasing and decreasing, expecting 3 intervals. if there is only one critical point, i can see two intervals but not three. am i missing a critical point here? i have excluded x = 2.645 because it is not a part of the domain.

 

[ehild]

i have excluded x = 2.645 because it is not a part of the domain.

It is a vertical asymptote there, and the derivative might change sign.

ehild

 

[shocklightnin]

but if i include it it shows that the function is increasing over intervals (-infinity,-0.345) U (2.645,+infinity) and decreasing on (-0.345,2.645). however i know that -0.345 is a relative minimum, and if those intervals hold, it becomes a relative maximum...

 

[ehild]

Check the sign of the derivative.

ehild

 

[shocklightnin]

i already know the answer is that the function f is decreasing on (-infty, -0.345 ) U (2.645 ,+infty ) and increasing on ( -0.345 , 2.645).
I am just not sure at how they arrived to it in my profs notes >.<

we do the table method where its like:

-0.345 2.645
------------------------------------
-1.38-4x | - 0 + | +
(-2x-5.29)^3 | - | - 0 +
f'(x) | + | - | +
f(x) | go up | go dwn| go up

my final f(x) ends up being the opposite of what its supposed to be :s I am not sure what I am doing wrong here. any help is much appreciated.

 

[ehild]

f'(x) = (-1.38-4x)/(2x-5.29)^3

You know that the function is increasing when its derivative is positive.

Is f' positive or negative, if x>2.645? Say, x=3. Is -1.38-4x negative or positive? Is 2x-5.29 negative or positive?

ehild

 

[shocklightnin]

Ohhh just got the mistake. Great, thank you for your help and patience!

 

[ehild]

You are welcome.

ehild