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

Section 4.3 How Derivatives Affect the Shape of a Graph 295
SOLUtion From the chart in the solution to Example 1 we see that f 9sxd changes from
negative to positive at 21, so f s21d − 0 is a local minimum value by the First Derivative
Test. Similarly, f 9 changes from negative to positive at 2, so f s2d − 227 is also a
local minimum value. As noted previously, f s0d − 5 is a local maximum value because
f 9sxd changes from positive to negative at 0.
n
ExamplE 3 Find the local maximum and minimum values of the function
tsxd − x 1 2 sin x
0 < x < 2
SOLUtion As in Example 1, we start by finding the critical numbers. The derivative is:
t9sxd − 1 1 2 cos x
so t9sxd − 0 when cos x − 2 1 2 . The solutions of this equation are 2y3 and 4y3.
Because t is differentiable everywhere, the only critical numbers are 2y3 and 4y3.
We split the domain into intervals according to the critical numbers. Within each
interval, t9sxd is either always positive or always negative and so we analyze t in the
following chart.
The 1 signs in the chart come from the
fact that t9sxd . 0 when cos x . 2 1 2 .
From the graph of y − cos x, this is
true in the indicated intervals.
Interval t9sxd − 1 1 2 cos x t
0 , x , 2y3 1 increasing on s0, 2y3d
2y3 , x , 4y3 2 decreasing on s2y3, 4y3d
4y3 , x , 2 1 increasing on s4y3, 2d
6
0 2π 4π
2π
3 3
FIGURE 4
tsxd − x 1 2 sin x
Because t9sxd changes from positive to negative at 2y3, the First Derivative Test tells
us that there is a local maximum at 2y3 and the local maximum value is
ts2y3d − 2 3 1 2 sin 2 3 − 2 3 1 2 S s3
2
D − 2 3 1 s3 < 3.83
Likewise, t9sxd changes from negative to positive at 4y3 and so
ts4y3d − 4 3 1 2 sin 4 3 − 4 3 1 2 S2 s3
2
D − 4 3 2 s3 < 2.46
is a local minimum value. The graph of t in Figure 4 supports our conclusion.
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What Does f 99 Say About f ?
Figure 5 shows the graphs of two increasing functions on sa, bd. Both graphs join point
A to point B but they look different because they bend in different directions. How can
we dis tinguish between these two types of behavior?
y
B
y
B
f
g
A
A
0 a
b x
0 a
b
x
FIGURE 5
(a)
(b)
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