Note that what is shown is the graph of the derivative
,
not the
graph of
!!
Critical points at
and
. Neither local max nor min at
;
neither local max nor min at
; local maximum at
.
Critical points at
and
. Local minimum at
; local maximum at
.
Critical points at
, and
. Neither local maximum nor minimum at
; local maximum at
.
Critical points at
, and
. Local minimum at
; local
maximum at
; neither local maximum or minimum at
.
Correct!
Correct!
Critical points at
and
. Local maximum at
; neither
local min nor local max at
;
local minimum at
.

The given graph is the graph of the derivative
. We see that
at
and
. So these are the three critical
points. We use the first derivative test to tell where
has a
local minimum, a local maximum, or neither at each point.
At
,
changes from negative to positive, so
a local minimum at
At
,
changes from positive to negative so
has a local maximum at
At
,
does not change sign (it is negative on either
side of
), so
has neither a local max nor a local min at
.
has
.
.

2.5 / 2.5 pts
Question 8
Suppose
is a continuous function that has critical points at
and at
such that
and
. The second derivative of
given as
. Use the second derivative test and
choose the correct statement regarding local
extrema at the given critical
points.
is
.

has a local min at
; the second derivative test tells us nothing about
what happens at
.
Correct!
Correct!
has a local max at
;
has neither a local max nor a local min at
.
has a local min at
and a local max at
.

has a local max at
; the second derivative test tells us nothing about
what happens at
.
If
is positive at a critical point
, then
has a local
minimum at
.
If
is negative at a critical point
, then
has a local
maximum at
.
If
at a critical point
, then the second derivative
test tells us nothing about what happens at
.
, so the second derivative test tells us nothing about
what happens at
.
, so
has a local minimum at
.