126 STEP 4. Review the Knowledge You Need to Score High
(c) The function f is increasing on (−∞,−4], [2, 4], and [8,∞) and is decreasing on
[−4, 2] and [4, 8].
(d) Summarize the information of f′′on a number line:A change of concavity occurs atx=−1, 3, and 6. Sincef′(x) exists, f has a tangent
at every point. Therefore,f has a point of inflection atx=−1, 3, and 6.
(e) The function f is concave upward on (−1, 3) and (6,∞) and concave downward on
(−∞,−1) and (3, 6).Example 4
A function f is continuous on the interval [−4, 3] with f(−4)=6 and f(3)=2 and the
following properties:INTERVALS (−4,−2) x=−2(−2, 1) x= 1 (1, 3)
f′ − 0 − undefined +
f′′ + 0 − undefined −(a) Find the intervals on which fis increasing or decreasing.
(b) Find where fhas its absolute extrema.
(c) Find where fhas the points of inflection.
(d) Find the intervals where fis concave upward or downward.
(e) Sketch a possible graph off.Solution:
(a) The graph of f is increasing on [1, 3] since f′>0 and decreasing on [−4,−2] and
[−2, 1] since f′<0.
(b) Atx=−4, f(x)=6. The function decreases untilx=1 and increases back to 2 at
x=3. Thus, f has its absolute maximum atx=−4 and its absolute minimum at
x=1.
(c) A change of concavity occurs atx=−2, and sincef′(−2)=0, which implies a tangent
line exists atx=−2,fhas a point of inflection atx=−2.
(d) The graph of f is concave upward on (−4, −2) and concave downward on
(−2, 1) and (1, 3).
(e) A possible sketch of fis shown in Figure 7.4-5.