Using the derivative of f(x) given below, determine the intervals on which f(x) is increasing or decreasing.
-$$f ^ { \prime } ( x ) = ( 1 - x ) ( 9 - x )$$
A) Decreasing on $( - \infty , - 1 ) \cup ( - 9 , \infty )$; increasing on $( - 1 , - 9 )$
B) Decreasing on $( - \infty , 1 ) \cup ( 9 , \infty )$; increasing on $( 1,9 )$
C) Decreasing on $( 1,9 )$; increasing on $( - \infty , 1 ) \cup ( 9 , \infty )$
D) Decreasing on $( - \infty , 1 )$; increasing on $( 9 , \infty )$

Question

Quizplus · Accepted Answer

To find intervals of increase and decrease, we need to look at the sign of the derivative. $ f'(x)=(1-x)(9-x) $ This is equal to zero when either $(1-x)$ or $(9-x)$ is zero, i.e., when $x=1$ or $x=9$. For $x<1$, both factors are positive, so $f'(x)>0$, meaning that $f(x)$ is increasing. For $19$, both factors are negative, so $f'(x)>0$, meaning that $f(x)$ is increasing. Combining these intervals gives the answer of C: Decreasing on $(1,9)$; increasing on $(-\infty,1)\cup(9,\infty)$.

Using the Derivative of F(x) Given Below, Determine the Intervals f′(x)=(1−x)(9−x)f ^ { \prime } ( x ) = ( 1 - x ) ( 9 - x )f′(x)=(1−x)(9−x)

Using the Derivative of F(x) Given Below, Determine the Intervals $f ^ { \prime } ( x ) = ( 1 - x ) ( 9 - x )$