Graphs of Functions and Derivatives 143
Step 7:Sketch the graph. (See
Figure 7.9-3.)
Figure 7.9-3
- Step 1: Domain: all real numbersx=/4.
Step 2:Symmetry: none.
Step 3:Find f′and f′′.
f′(x)=
( 1 )(x− 4 )−( 1 )(x+ 4 )
(x− 4 )^2
=
− 8
(x− 4 )^2
, f′′(x)=
16
(x− 4 )^3
Step 4:Critical numbers:f′(x)=/0 and
f′(x) is undefined atx=4.
Step 5:Determine intervals.
Intervals are (−∞, 4) and (4,∞).
Step 6:Set up table as below:
INTERVALS (−∞, 4) (4,∞)
f′ −−
f′′ −+
conclusion decr concave
downward
incr concave
upward
Step 7:Horizontal asymptote:
x→±∞lim
x+ 4
x− 4
=1. Thus,y=1isa
horizontal asymptote.
Vertical asymptote:
xlim→ 4 +
x+ 4
x− 4
=∞and
xlim→ 4 −
x+ 4
x− 4
=−∞. Thus,x=4isa
vertical asymptote.
Step 8: x-intercept: Set f′(x)= 0
⇒x+ 4 =0;x=−4.
y-intercept: Setx= 0
⇒f(x)=−1.
Step 9: Sketch the graph. (See
Figure 7.9-4.)
Figure 7.9-4
- (a)
The functionfhas the largest value
(of the four choices) atx=x 1. (See
Figure 7.9-5.)
Figure 7.9-5
