Answer
(a) The intervals when $f$ is increasing:
$(0,4)$
$(6,8)$
(b) The values when $f$ has a local maximum:
$x = 4$
$x = 8$
The value when $f$ has a local minimum:
$x = 6$
(c) The intervals when $f$ is concave upward:
$(0,1)$
$(2,3)$
$(5,7)$
The intervals when $f$ is concave downward:
$(1,2)$
$(3,5)$
$(7,9)$
(d) The inflection points are:
$x = 1$
$x = 2$
$x = 3$
$x = 5$
$x = 7$
Work Step by Step
(a) $f$ is increasing when $f' \gt 0$
The intervals when $f$ is increasing:
$(0,4)$
$(6,8)$
(b) $f$ has a local maximum when $f'$ changes from positive to negative.
$f$ has a local minimum when $f'$ changes from negative to positive.
The values when $f$ has a local maximum:
$x = 4$
$x = 8$
The value when $f$ has a local minimum:
$x = 6$
(c) $f$ is concave upward when $f'$ is increasing.
$f$ is concave downward when $f'$ is decreasing.
The intervals when $f$ is concave upward:
$(0,1)$
$(2,3)$
$(5,7)$
The intervals when $f$ is concave downward:
$(1,2)$
$(3,5)$
$(7,9)$
(d) The inflection points are the points where the $f'(x) = 0$ at a local maximum or a local minimum.
The inflection points are:
$x = 1$
$x = 2$
$x = 3$
$x = 5$
$x = 7$
