James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 4.3 How Derivatives Affect the Shape of a Graph 297
SOLUtion By looking at the slope of the curve as t increases, we see that the rate
of increase of the population is initially very small, then gets larger until it reaches a
maximum at about t − 12 weeks, and decreases as the population begins to level off.
As the population approaches its maximum value of about 75,000 (called the carrying
capacity), the rate of increase, P9std, approaches 0. The curve appears to be concave
upward on (0, 12) and concave downward on (12, 18).
n
In Example 4, the population curve changed from concave upward to concave downward
at approximately the point (12, 38,000). This point is called an inflection point of
the curve. The significance of this point is that the rate of population increase has its
maximum value there. In general, an inflection point is a point where a curve changes its
direction of concavity.
Definition A point P on a curve y − f sxd is called an inflection point if f is continuous
there and the curve changes from concave upward to concave downward or
from concave downward to concave upward at P.
For instance, in Figure 7, B, C, D, and P are the points of inflection. Notice that if a
curve has a tangent at a point of inflection, then the curve crosses its tangent there.
In view of the Concavity Test, there is a point of inflection at any point where the
second derivative changes sign.
ExamplE 5 Sketch a possible graph of a function f that satisfies the following
conditions:
y
sid f 9sxd . 0 on s2`, 1d,
f 9sxd , 0 on s1, `d
siid f 0sxd . 0 on s2`, 22d and s2, `d,
siiid lim f sxd − 22, lim f sxd − 0
x l2` x l`
f 0sxd , 0 on s22, 2d
-2
y=_2
FIGURE 9
0 1 2
x
SOLUtion Condition (i) tells us that f is increasing on s2`, 1d and decreasing on
s1, `d. Condition (ii) says that f is concave upward on s2`, 22d and s2, `d, and concave
downward on s22, 2d. From condition (iii) we know that the graph of f has two
horizontal asymptotes: y − 22 (to the left) and y − 0 (to the right).
We first draw the horizontal asymptote y − 22 as a dashed line (see Figure 9). We
then draw the graph of f approaching this asymptote at the far left, increasing to its
maximum point at x − 1, and decreasing toward the x-axis as at the far right. We also
make sure that the graph has inflection points when x − 22 and 2. Notice that we
made the curve bend upward for x , 22 and x . 2, and bend downward when x is
between 22 and 2.
n
y
f
Another application of the second derivative is the following test for identifying local
maximum and minimum values. It is a consequence of the Concavity Test, and serves as
an alternative to the First Derivative Test.
f ª(c)=0
P
f(c)
ƒ
The Second Derivative Test Suppose f 0 is continuous near c.
(a) If f 9scd − 0 and f 0scd . 0, then f has a local minimum at c.
(b) If f 9scd − 0 and f 0scd , 0, then f has a local maximum at c.
0
FIGURE 10
f 99scd . 0, f is concave upward
c
x
x
For instance, part (a) is true because f 0sxd . 0 near c and so f is concave upward
near c. This means that the graph of f lies above its horizontal tangent at c and so f has
a local minimum at c. (See Figure 10.)
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