Thomas Calculus 13th [Solutions]

Section 4.4 Concavity and Curve Sketching 269
77.
y
2/3
x ( x 1), y )( | rising on (1, ),
0 1
falling on ( ,1) no local maximum, but
2/3 5/3
a local minimum at x 1; y 1 x 2 x
3 3
1 5/3
x ( x 2), y | )( concave up on
3
2 0
( , 2) and (0, ), concave down on ( 2, 0) points of
inflection at x 2 and x 0, and a vertical tangent at x 0
78.
y
4/5
x ( x 1), y | ) ( rising on ( 1, 0) and
1 0
(0, ), falling on ( , 1) no
local maximum, but a local minimum at x 1;
y 1 4/5 4 9/5
x x 1 9/5
5 5 5 x ( x 4), y )( |
0 4
concave up on ( , 0) and (4, ), concave down on (0, 4)
points of inflection at x 0 and x 4, and a vertical tangent
at x 0
79.
2 x, x 0
y
, y | rising on ( , ) no
2 x, x 0
0
2, x 0
local extrema; y
, y )( concave up
2, x 0
0
on (0, ), concave down on ( , 0) a point of inflection at
x 0
80.
y
2
x , x 0 , y |
2
x , x 0
0
rising on (0,
( , 0) no local maximum,
but a local minimum at x 0; y
2 x, x 0
,
2 x, x 0
y | concave up on ( , )
0
no point of inflection
), falling on
81. The graph of y f ( x ) the graph of y f ( x ) is concave up on
(0, ), concave down on ( , 0) a point of inflection at x 0;
the graph of y f ( x ) y | | the graph
y f ( x ) has both a local maximum and a local minimum
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