Where f ′ ( x ) > 0 {\displaystyle \displaystyle f'(x)>0} , f ( x ) {\displaystyle \displaystyle f(x)} is increasing; where f ′ ( x ) < 0 {\displaystyle \displaystyle f'(x)<0} , f ( x ) {\displaystyle \displaystyle f(x)} is decreasing.
From part (a), we note that f ′ ( x ) {\displaystyle \displaystyle f'(x)} may change sign at its critical points x = 0 {\displaystyle \displaystyle x=0} and x = 1 {\displaystyle \displaystyle x=1} . We can construct a sign table to organize our calculations:
From this table we observe that f ( x ) {\displaystyle \displaystyle f(x)} is increasing on x ∈ ( 0 , 1 ) {\displaystyle {\color {blue}x\in (0,1)}} and decreasing on x ∈ ( − ∞ , 0 ) ∪ ( 1 , ∞ ) {\displaystyle {\color {OliveGreen}x\in (-\infty ,0)\cup (1,\infty )}} .
, 
is increasing; where 
, is decreasing.
may change sign at its critical points 
and 
. We can construct a sign table to organize our calculations:
is increasing on 
and decreasing on 
.
