Chapter 5: Applications of Differentiation
Section 5.1: Extrema and the Mean Value Theorem
Definition: Let f be a function with domain D that contains the number c.
• The function has an absolute maximum at x = c if f (c) ≥ f (x) for all x ∈ D. The
number f (c) is called the absolute maximum value of f on D. Similarly, f has an
absolute minimum at c if f (c) ≤ f (x) for all x ∈ D and the number f (c) is called
the absolute minimum value of f on D. The absolute maximum and minimum
values of f are called the absolute extrema of f .
• The function has a local maximum at x = c if f (c) ≥ f (x) for all x in some open
interval containing c. The number f (c) is called a local maximum value of f .
Similarly, f has a local minimum at c if f (c) ≤ f (x) for all x in some open interval
containing c and the number f (c) is called a local minimum value of f . The local
maximum and minimum values of f are called the local extrema of f .
Example: Find the absolute and local extrema of each function by graphing.
(a) f (x) = 1 − x2 , −2 ≤ x ≤ 1
1
(b) f (x) =
1
,0<x≤1
x
(c) f (x) = cos x, −2π ≤ x ≤ 2π
(d) f (x) = e−|x| , −1 ≤ x ≤ 1
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Theorem: (Fermat’s Theorem)
If f has a local maximum or minimum at x = c and f 0 (c) exists, then f 0 (c) = 0.
Note: If f 0 (c) = 0, f does not necessarily have an extreme value at c. For instance, if
f (x) = x3 , then f 0 (0) = 0 but f has no maximum or minimum at x = 0. Similarly, if f
has an extreme value at c, then f 0 may not exist. For example, f (x) = |x| has an absolute
minimum at x = 0, but f 0 (0) does not exist.
Definition: A critical number of a function f is a number c in the domain of f such that
either f 0 (c) = 0 or f 0 (c) does not exist.
Example: Find the critical numbers of each function.
(a) f (x) = 4x3 − 9x2 − 12x + 3
3
(b) g(x) = |4 − x2 |
(c) f (x) =
(d) g(x) =
x
x2 + 1
√
x2 − x
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Theorem: (The Extreme Value Theorem)
If f is continuous on a closed interval [a, b], then f attains an absolute maximum value and
absolute minimum value on [a, b].
The Closed Interval Method: To find the absolute extrema of a continuous function f on a
closed interval [a, b]:
1. Find the values of f at each of the critical numbers in (a, b).
2. Find the values of f at the endpoints x = a and x = b.
3. The largest of these values is the absolute maximum and the smallest is the absolute
minimum.
Example: Find the absolute extrema of each function on the given interval.
(a) f (x) = x3 − 12x + 1, [−3, 5]
(b) g(x) = 3x5 − 5x3 − 1, [−2, 2]
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(c) f (x) = x − 2 cos x, [0, 2π]
(d) g(x) = xe−x , [0, 1]
Example: (Logistic Growth) Suppose that the size of a population at time t ≥ 0 is N (t) and
the growth rate of the population is given by
dN
N
= rN 1 −
dt
K
where r and K are positive constants. Find the population size for which the growth rate is
maximal.
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Theorem: (Mean Value Theorem)
If f is continuous on [a, b] and differentiable on (a, b), then there exists at least one number
c ∈ (a, b) such that
f (b) − f (a)
f 0 (c) =
.
b−a
Example: Verify that the function f (x) = x3 + x − 1 satisfies the hypotheses of the Mean
Value Theorem on [0, 2] and find the value c that satisfies the theorem.
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Example: (Population Growth) Let N (t) denote the size of a population at time t ≥ 0. If
the initial population is N (0) = 50 and the growth rate satisfies |N 0 (t)| ≤ 2 for all t ∈ [0, 5].
What can you say about N (5)?
Example: A car moves in a straight line according to the position function
s(t) =
1 2
t,
10
0 ≤ t ≤ 10
where s is measured in meters and t is measured in seconds.
(a) Find its average velocity between t = 0 and t = 10.
(b) Find its instantaneous velocity at time t.
(c) At what time does the instantaneous velocity equal the average velocity?
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