Graphs of Functions and Derivatives 145Sincef′(3) exists and a change of
concavity occurs atx=3,fhas a
point of inflection atx=3.
(d) Concave upward on (−∞, 3) and
downward on (3,∞).
(e)Sketch a graph. (See Figure 7.9-7.)Figure 7.9-7- (See Figure 7.9-8.)
[–5,10] by [–5,20]
Figure 7.9-8The graph offindicates that a relative
maximum occurs atx=3,fis not
differentiable atx=7, since there is a cusp
atx=7, and fdoes not have a point of
inflection atx=−1, since there is no
tangent line atx=−1. Thus, only
statementIis true.- (See Figure 7.9-9.)
[–π,π] by [–2,2]
Figure 7.9-9Entery 1 =cos(x^2 )
Using the [Inflection] function of your
calculator, you obtain three points of
inflection on [0,π]. The points of
inflection occur atx= 1 .35521, 2.1945,
and 2.81373. Sincey 1 =cos(x^2 ) is an even
function, there is a total of 6 points of
inflection on [−π,π]. An alternate
solution is to enter
y 2 =
d^2
dx^2(
y 1 (x),x,2). The graph ofy 2
indicates that there are 6 zeros on [−π,π].
19. Entery 1 = 3 ∗e∧(−x∧ 2 /2). Note that
the graph has a symmetry about they-axis.
Using the functions of the calculator, you
will find:
(a) a relative maximum point at (0, 3),
which is also the absolute maximum
point;
(b) points of inflection at (−1, 1.819) and
(1, 1.819);
(c) y=0 (thex-axis) a horizontal
asymptote;
(d) y 1 increasing on (−∞, 0] and
decreasing on [0,∞); and
(e) y 1 concave upward on (−∞,−1) and
(1,∞) and concave downward on
(−1, 1). (See Figure 7.9-10.)