Engineering Calculus
posted by Cheallaigh .
f(x)=((6x3)/(x+6)) how do you find the intervals of decreasing and increasing...I found that there were no critical points yet I was also correct that the function increased (I,6) and (6,I). I was wrong that there were no decreasing intervals though.

If fŒ(x) > 0 at each point in an interval I, then the function is said to be increasing on I.
fŒ(x) < 0 at each point in an interval I, then the function is said to be decreasing on I.
In your case :
fŒ(x) = 39 /( x + 6 ) ^ 2
39 positive
( x + 6 ) ^ 2 always positive except
when x =  6
In point x =  6 function has vertical asymptote
So function :
f( x ) = ( 6 x  3 ) / ( x + 6)
always increasing
P.S.
If you don't know how to find first derivation
Go on:
wolframalpha dot com
When page be open in rectangle type:
derivative (6x3)/(x+6)
and click option =
When you see result click option:
Show steps
If you want to see graph of your function in google type:
function graphs online
When you see list of results click on:
rechneronline.de/functiongraphs
When page be open in blue rectacangle type:
(6x3)/(x+6)
Set :
Range xaxis from 100 to 100
Range yaxis from 100 to 100
Then click option :
Draw
You will see graph of your function.