Question 1110853: Find an interval over which the function f(x)=(x+5)^3(x-4)^2 is decreasing.
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! The interval over which the function  is decreasing
is the interval where its derivative is negative.
 can be considered to be the product of the functions
 and 
You must have learned that for a product function  ,
the derivative  can be calculated
from the factor functions,  and  ,
and their derivatives,  and  , as
 .
In this case,  and  , so
The zeros of that derivative are  ,  and  .
Because has even multiplicity
(multiplicity is 2 because ( is squared).
 does not change sign at  .
the derivative changes sign at the other two roots,
and it is obviously positive for  ,
so it is negative only in the interval  .
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