3.1. USING DERIVATIVES TO IDENTIFY EXTREME VALUES
163
-2
-1
1
2
-2
-1
1
2
y = h(x)
Figure 3.2: The graph of a function h on the interval [−3, 3].
(c) Does h have a global maximum on the interval [3, 3]? If so, what is the value of
this global maximum?
(d) Does h have a global minimum on the interval [3, 3]? If so, what is its value?
(e) Identify all values of c for which h ′ (c) = 0.
(f) Identify all values of c for which h ′ (c) does not exist.
(g) True or false: every relative maximum and minimum of h occurs at a point where
h ′ (c) is either zero or does not exist.
(h) True or false: at every point where h ′ (c) is zero or does not exist, h has a relative
maximum or minimum.
⊲⊳
Critical numbers and the first derivative test
If a function has a relative extreme value at a point (c, f (c)), the function must change its
behavior at c regarding whether it is increasing or decreasing before or after the point.
For example, if a continuous function has a relative maximum at c, such as those
pictured in the two leftmost functions in Figure 3.3, then it is both necessary and sufficient
that the function change from being increasing just before c to decreasing just after c. In
the same way, a continuous function has a relative minimum at c if and only if the function
changes from decreasing to increasing at c. See, for instance, the two functions pictured
at right in Figure 3.3. There are only two possible ways for these changes in behavior to
occur: either f ′ (c) = 0 or f ′ (c) is undefined.
163
-2
-1
1
2
-2
-1
1
2
y = h(x)
Figure 3.2: The graph of a function h on the interval [−3, 3].
(c) Does h have a global maximum on the interval [3, 3]? If so, what is the value of
this global maximum?
(d) Does h have a global minimum on the interval [3, 3]? If so, what is its value?
(e) Identify all values of c for which h ′ (c) = 0.
(f) Identify all values of c for which h ′ (c) does not exist.
(g) True or false: every relative maximum and minimum of h occurs at a point where
h ′ (c) is either zero or does not exist.
(h) True or false: at every point where h ′ (c) is zero or does not exist, h has a relative
maximum or minimum.
⊲⊳
Critical numbers and the first derivative test
If a function has a relative extreme value at a point (c, f (c)), the function must change its
behavior at c regarding whether it is increasing or decreasing before or after the point.
For example, if a continuous function has a relative maximum at c, such as those
pictured in the two leftmost functions in Figure 3.3, then it is both necessary and sufficient
that the function change from being increasing just before c to decreasing just after c. In
the same way, a continuous function has a relative minimum at c if and only if the function
changes from decreasing to increasing at c. See, for instance, the two functions pictured
at right in Figure 3.3. There are only two possible ways for these changes in behavior to
occur: either f ′ (c) = 0 or f ′ (c) is undefined.