Thomas Calculus 13th [Solutions]

270 Chapter 4 Applications of Derivatives
82. The graph of y f ( x) y | the graph of
y f ( x ) has a point of inflection, the graph of
y f ( x) y | | the graph of y f ( x ) has
both a local maximum and a local minimum
83. The graph of y f ( x) y | |
the graph of y f ( x ) has two points of inflection, the graph of
y f ( x)
y | the graph of y f ( x ) has a local
minimum
84. The graph of y f ( x) y | the graph of
y f ( x ) has a point of inflection; the graph of
y f ( x) y | | the graph of y f ( x ) has
both a local maximum and a local minimum
85.
y
2
2x
x 1
2
x 1
Since 1 and 1 are roots of the denominator, the domain is
( , 1) ( 1,1) (1, ).
1 2
y ; y ( x 1)
2 3
( x 1) ( x 1)
There are no critical points. The function is decreasing on its
domain. There are no inflection points. The function is concave
down on ( , 1) ( 1,1) and concave up on (1, ). The
numerator and denominator share a factor of x 1. Dividing out
this common factor gives y 2x
1
x 1 ( x 1), which shows that
x 1 is a vertical asymptote. Now dividing numerator and
2 (1/ x)
denominator by x gives y
1 (1/ x) , which shows that y 2 is a
horizontal asymptote. The graph will have a hole at x 1,
y 2( 1) 1 3
1( 1) 1 2 . The x-intercept is 1 2 .
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