Question

(i) identify the critical numbers of ( f ). (enter your answers as a comma - separated list.)( x = )
(ii) identify the open interval(s) on which ( f ) is increasing or decreasing. (enter your answer using interval notation.)
increasing
decreasing
(iii) determine whether ( f ) has a relative maximum, a relative minimum, or neither at each critical number. (enter your answers as comma - separated lists.)
relative maximum ( x = )
relative minimum ( x = )

Explanation:

Step1: Analyze critical x=0

At $x=0$, the function changes from increasing ($(-\infty,0)$) to decreasing ($(0,1)$). This means the function reaches a peak here, so it is a relative maximum.

Step2: Analyze critical x=1

At $x=1$, the function changes from decreasing ($(0,1)$) to increasing ($(1,2)$). This means the function reaches a valley here, so it is a relative minimum.

Step3: Analyze critical x=2

At $x=2$, the function changes from increasing ($(1,2)$) to decreasing ($(2,\infty)$). This means the function reaches a peak here, so it is a relative maximum.

Answer:

relative minimum $x = 1$
relative maximum $x = 0, 2$