MA 3972-MA-Book April 11, 2018 17:21
Graphs of Functions and Derivatives 155
Step 4: Critical numbers:
f′(x)= 0 ⇒− 42 x=0orx= 0
f′(x) is undefined atx=±5 which are not in the domain.
Possible points of inflection:
f′′(x)=0 andf′′(x) is undefined atx=±5 which are not in the domain.
Step 5: Determine the intervals:
- 550
Intervals are (−∞,−5), (−5, 0), (0, 5), and (5,∞).
Step 6: Set up a table:
INTERVALS (−∞,−5) X=−5(−5, 0) X= 0 (0, 5) X= 5 (5,∞)
f(x) undefined 4/25 undefined
f′(x) + undefined + 0 − undefined −
f′′(x) + undefined −−−undefined +
incr. incr. decr. decr.
concave concave concave concave
conclusion upward downward rel. max. downward upward
Step 7: Vertical asymptote:x=5 andx=− 5
Horizontal asymptote:y= 1
Step 8: y-intercept:
(
0,
4
25
)
x-intercept: (−2, 0) and (2, 0)
(See Figure 8.3-1.)
[–8, 8] by [–4, 4]
Figure 8.3-1
Graphing with Calculators
Example 1
Using a calculator, sketch the graph off(x)=−x^5 /^3 + 3 x^2 /^3 indicating all relative extrema,
points of inflection, horizontal and vertical asymptotes, intervals wheref(x) is increasing
or decreasing, and intervals wheref(x) is concave upward or downward.