PRACTICE TEST 4 SOLUTIONS
1.
Find intervals on which f is increasing, decreasing; open intervals where f is concave up, concave
down; and x coordinates of all inflection points; also state which points are stationary points,
PONDS, and classify as relative maximum or minimum points.
(
2
3
2
2
f
x
x
x
x
=

+

IT’S A
POLYNOMIAL – SO NO PONDS!
(
(
(
(
2
2
1
4
3
3
4
1
3
1
1
f
x
x
x
x
x
x
x
′
=  +

= 

+
= 


so there are 2 stationary points, at
x = 1 and x = 1/3
Sign graph shows
f
′
< 0 for x < 1/3 and x > 1;
f
′
> 0 for 1/3 < x < 1; hence
there’s a rel min at x = 1/3 and rel max at x = 1
(
(
4
6
6 2/3
f
x
x
x
′′
=

=

0
f
′′
if x < 2/3,
f
′
< 0 for x > 2/3 and an inflection point at x = 2/3
2.
GRAPH
Find and show intercepts vertical and horizontal asymptotes and SHOW SIGN
GRAPHS FOR
f
′
AND
f
′′
:
(
29
2
4
3
x
f
x
x

=

GIVEN:
(
29
(
29
2
2
;
3
f
x
x
′
=

(
29
(
29
3
4
3
f
x
x
′′
=

(2, 0)
and (0,  4/3) are intercepts;
x = 3 is a vertical asymptote and y = 2 is a horizontal
asymptote;
f
′
> 0 [except at x = 3], so
f
is always increasing [and no sign graph needed for
f
′
]
0
f
′′<
if x < 3 [hence f concave down];
0
f
′′
if x > 3 [hence f concave up] Note: the vertical
asmyptote should be drawn as a broken line – but I can’t show it here.
3.
Find the absolute maximum and minimum values of
(
29
[
]
1
sin
on 0,
2
f
x
x
x
π
=
+
Since this is on a closed interval, the Extreme Value Theorem lets us find critical points then
just evaluate f at the end points and critical point[s]
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 Fall '08
 STAFF
 Calculus, Critical Point, Inflection Points, lim, Mathematical analysis

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