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TEST 4 SOLUTIONS
1.[15] Find intercepts. intervals on which f is increasing, decreasing; open intervals where f is concave
up, concave
down; and xcoordinates of all inflection points;
also state which points [x
coordinates only] are
stationary points, PONDS, and classify as relative maximum or minimum
points; specify equation of any vertical tangent.
DO NOT SKETCH
( 29
4/3
1/3
f x
x
x
=

[SHOW SIGN GRAPHS AS USED IN CLASS FOR BOTH
f and f
′
′′
]
( 29 ( 29
4/3
1/3
1/3
1
f x
x
x
x
x
=

=

so the intercepts are (0,0) and (1,0)
( 29
1/3
2/3
1/3
2/3
2/3
4
1
1
1
4
1
4
3
3
3
3
x
f
x
x
x
x
x
x


′
=

=

=
so there is a POND at x = 0 and stationary point
at x = ¼ and the the sign graph for
f
′
shows
f
is increasing on x > ¼, decreasing on x < ¼
[the denominator is always > 0] x = 0 has a vertical tangent there [the derivative doesn’t change
sign there] so I guess I mislead you about that – sorry.
x = ¼ is a minimum. [No vertical
( 29
2/3
5/3
2/3
5/3
5/3
4
2
2
2
1
2
1
9
9
9
9
x
f
x
x
x
x
x
x


+
′′
=
+
=
+
=
and this changes sign at both x = 0 and x =  ½
So both these points are inflection points [since f is defined at both]
A sign graph of
f
′′
shows that f is concave down for 1/2 < x < 0 [
f
′′
< 0 there]
and concave up
for x <  ½ and x > 0 [
f
′′
> 0 for both of these]
2.[10]Find the absolute maximum and minimum values of
( 29 [ ]
cos on
/ 2,
f x
x
x
π
=
+

Since f is continuous everywhere, the Extreme Value Theorem applies and we are
guaranteed both an absolute maximum and an absolute minimum on the closed interval.
All
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 Fall '08
 STAFF
 Calculus, Inflection Points

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