Answers and Explanations 291
Answers
401–500
448.
absolute maximum: y = 3; no absolute minimum; local maxima: (3, 3), (5, 3); local
minimum: (4, 1)
The absolute maximum value is 3, which the graph attains at the points (3, 3) and
(5, 3). The function approaches the x-axis but doesn’t cross or touch it, so there’s no
absolute minimum. The points (3, 3) and (5, 3) also correspond to local maxima, and
the local minimum occurs at (4, 1).
449.
no maxima or minima
The graph has no absolute maximum or minimum because the graph approaches ∞
on the left and –∞ on the right. Likewise, there are no local maxima or minima because
(2, 1) and (5, 3) are points of discontinuity.
450.
I and II
Because Point A satisfies the definition of both a local maximum and a local minimum,
it’s both. The graph decreases to negative infinity on the left side, so Point A is not an
absolute minimum.
451.
absolute maximum: 5; absolute minimum: –7
Begin by finding the derivative of the function; then find any critical numbers on the
given interval by determining where the derivative equals zero or is undefined. Note
that by finding critical numbers, you’re finding potential turning points or cusp points
of the graph.
The derivative of f
 
(x) = 3x
2
– 12x + 5 is
′
f x
x
x
( )
(
)
=
−
=
−
6 12
6
2
Setting the derivative equal to zero and solving gives you the only critical number,
x = 2. Next, substitute the endpoints of the interval and the critical number into the
original function and pick the largest and smallest values:
f
f
f
( )
( )
( )
( )
( )
( )
( )
( )
( )
0 3 0
12 0 5 5
2 3 2
12 2 5
7
3
3 3
12 3
2
2
2
=
−
+ =
=
−
+ = −
=
−
+ 5 5
4
= −
Therefore, the absolute maximum is 5, and the absolute minimum is –7.
452.
absolute maximum: 67; absolute minimum: 3
Begin by finding the derivative of the function; then find any critical numbers on the
given interval. The derivative of f
 
(x) = x
4
– 2x
2
+ 4 is
′ ( )
( )
f x
x
x
x x
=
−
=
−
4
4
4
1
3
2