Math 1520 Course Notes Sections: 6.1, 6.2

Section 6.1: Absolute Extrema
Section 6.2: Applications of Extrema
Math 1520
Course Notes
Sections: 6.1, 6.2
Liliana Menjivar
Liliana Menjivar
Math 1520 Course Notes Sections: 6.1, 6.2
Section 6.1: Absolute Extrema
Section 6.2: Applications of Extrema
Section 6.1: Absolute Extrema
absolute max or min
extreme value theorem
closed interval method
first derivative test for absolute extreme values
second derivative test for absolute extreme values
Section 6.2: Applications of Extrema
guidelines
applications
Liliana Menjivar
Math 1520 Course Notes Sections: 6.1, 6.2
Section 6.1: Absolute Extrema
Section 6.2: Applications of Extrema
absolute max or min
The largest possible value of a function is called the absolute
maximum. The smallest possible value of a function is called the
absolute minimum.
Liliana Menjivar
Math 1520 Course Notes Sections: 6.1, 6.2
Section 6.1: Absolute Extrema
Section 6.2: Applications of Extrema
absolute max or min
Definition (Global Max/Min)
Suppose f (x) is defined on some interval I . Let c be a number in
I . Then
I
f (c) is the absolute (or global) maximum of f on I if
f (c) ≥ f (x),
I
∀x in I .
f (c) is the absolute minimum of f on I if
f (c) ≤ f (x),
∀x in I .
A function has a absolute extremum (plural: extrema) of f (c) if it
has either an absolute max or absolute min there.
Liliana Menjivar
Math 1520 Course Notes Sections: 6.1, 6.2
Section 6.1: Absolute Extrema
Section 6.2: Applications of Extrema
extreme value theorem
Theorem (Extreme Value Theorem(EVT))
If function f is continuous on a closed interval [a, b], then f attains
both an absolute maximum and an absolute minimum on [a, b].
The EVT guarantees the existence of absolute extrema for a
continuous function defined on a closed interval.
Liliana Menjivar
Math 1520 Course Notes Sections: 6.1, 6.2
Section 6.1: Absolute Extrema
extreme value theorem
Liliana Menjivar
Math 1520 Course Notes Sections: 6.1, 6.2
Section 6.2: Applications of Extrema
Section 6.1: Absolute Extrema
Section 6.2: Applications of Extrema
extreme value theorem
What if the hypotheses of the EVT fail? That is, what if f is
discontinuous? what if f is not defined on a closed interval
Liliana Menjivar
Math 1520 Course Notes Sections: 6.1, 6.2
Section 6.1: Absolute Extrema
extreme value theorem
Liliana Menjivar
Math 1520 Course Notes Sections: 6.1, 6.2
Section 6.2: Applications of Extrema
Section 6.1: Absolute Extrema
Section 6.2: Applications of Extrema
closed interval method
To find absolute extrema for a continuous function f on [a, b] we
use the closed interval method (CIM).
1. Find all critical numbers of f in (a, b) .
2. Evaluate f at its critical number(s) and at the endpoints a
and b. That is, find f (c), f (a), and f (b).
3. The absolute maximum of f on [a, b] is the largest value found
in step 2, and the absolute minimum of f on [a, b] is the
smallest value found in step 2.
Liliana Menjivar
Math 1520 Course Notes Sections: 6.1, 6.2
Section 6.1: Absolute Extrema
Section 6.2: Applications of Extrema
closed interval method
Example 1. Find the absolute maximum and minimum of
f (x) = x 3 − 3x 2 + 1,
Liliana Menjivar
Math 1520 Course Notes Sections: 6.1, 6.2
1
[− , 4].
2
Section 6.1: Absolute Extrema
Section 6.2: Applications of Extrema
closed interval method
Example 2. Find the absolute maximum and minimum of the
average cost for the given cost function:
C (x) =
Liliana Menjivar
Math 1520 Course Notes Sections: 6.1, 6.2
x 2 + 36
,
2
[1, 12].
Section 6.1: Absolute Extrema
Section 6.2: Applications of Extrema
closed interval method
Example 3. From information given in a recent business
publication, a mathematical model was constructed to represent
the miles per gallon used by a certain car at a speed of x mph:
1 2
x + 2x − 20, 30 ≤ x ≤ 65.
45
Find the absolute maximum miles per gallon and the absolute
minimum miles per gallon and the speeds at which they occur.
M(x) = −
Liliana Menjivar
Math 1520 Course Notes Sections: 6.1, 6.2
Section 6.1: Absolute Extrema
Section 6.2: Applications of Extrema
first derivative test for absolute extreme values
Theorem (The First Derivative Test for Absolute Extreme
Values)
Suppose f is continuous on an interval I and that f has exactly
one critical number x = c in I ,
0
0
0
0
I
If f (x) > 0 on ∀x < c and f (x) < 0 ∀x > c, then f (c) is the
absolute maximum of f .
I
If f (x) < 0 on ∀x < c and f (x) > 0 ∀x > c, then f (c) is the
absolute minimum of f .
Liliana Menjivar
Math 1520 Course Notes Sections: 6.1, 6.2
Section 6.1: Absolute Extrema
Section 6.2: Applications of Extrema
second derivative test for absolute extreme values
Theorem (The Second Derivative Test for Absolute Extreme
Values)
Suppose f is continuous on an interval I and that f has exactly
one critical number x = c in I ,
00
I
If f (x) < 0 ∀x ∈ I , then f (c) is the absolute maximum of f .
I
If f (x) > 0 ∀x ∈ I , then f (c) is the absolute minimum of f .
00
Liliana Menjivar
Math 1520 Course Notes Sections: 6.1, 6.2
Section 6.1: Absolute Extrema
Section 6.2: Applications of Extrema
guidelines
(1) Understand the problem.
(a) Determine what is given and what is unknown.
(b) Determine the quantity that is being maximized or minimized.
(2) Draw a diagram.
(3) Introduce notation.
(a) Create variables for all the known and unknown quantities.
(b) Create a variable for the quantity being maximized or
minimized (for now call it Q).
Liliana Menjivar
Math 1520 Course Notes Sections: 6.1, 6.2
Section 6.1: Absolute Extrema
Section 6.2: Applications of Extrema
guidelines
(4) Find any relationships between the variables in step 3.
(5) Express Q as a function of all other variables and constant.
Then, using step 4 express Q as a function of only one
variable and state the domain of Q.
Liliana Menjivar
Math 1520 Course Notes Sections: 6.1, 6.2
Section 6.1: Absolute Extrema
Section 6.2: Applications of Extrema
guidelines
(6) Find the absolute maximum or minimum values of Q.
(a) If f continuous and D = [a, b] use CIM.
(b) If f has one critical number and D = I use the first derivative
test for absolute extreme values.
(c) If f has one critical number and D = I use the second
derivative test for absolute extreme values.
(d) If f has more than one CN and D = I , find any local extrema
and then calculate the limit of f at the endpoints of D to
determine if the local extrema are absolute extrema.
Liliana Menjivar
Math 1520 Course Notes Sections: 6.1, 6.2
Section 6.1: Absolute Extrema
Section 6.2: Applications of Extrema
applications
Example 1. A farmer has 500 meters of fencing with which to
fence in three sides of a rectangular pasture. A straight river will
form the fourth side. Find the dimensions of the pasture of the
largest area that the farmer can fence.
Liliana Menjivar
Math 1520 Course Notes Sections: 6.1, 6.2
Section 6.1: Absolute Extrema
Section 6.2: Applications of Extrema
applications
Example 2. A theater owner charges $5.00 per ticket and sells
250 tickets. By checking other theaters, the owner decides that for
every one dollar he raises the ticket price, he will lose 10
customers. What should he charge to maximize revenue?
Liliana Menjivar
Math 1520 Course Notes Sections: 6.1, 6.2
Section 6.1: Absolute Extrema
Section 6.2: Applications of Extrema
applications
Example 3. A company wants to manufacture cylindrical
aluminum cans that hold 1000 cm3 (1 liter) of oil. What should
the radius and height of the can be to minimize the amount of
aluminum used?
Liliana Menjivar
Math 1520 Course Notes Sections: 6.1, 6.2
Section 6.1: Absolute Extrema
Section 6.2: Applications of Extrema
applications
Example 4. A carpenter wants to make an open-topped box out
of a square piece of tin, 12 in wide, by cutting out a square from
each of the four corners and bending up the sides. What size
square should be cut from each corner to produce a box of
maximum volume?
Liliana Menjivar
Math 1520 Course Notes Sections: 6.1, 6.2
Section 6.1: Absolute Extrema
Section 6.2: Applications of Extrema
applications
Example 5. A company manufactures and sells x smart phones
per week. The weekly demand and cost equations are, respectively,
p = 500 − .5x,
C (x) = 20, 000 + 135x
(a) What price should the company charge for the phones, and
how many phones should be produced to maximize the weekly
revenue? What is the maximum weekly revenue?
(b) What is maximum weekly profit? How much should the
company charge for the phones, and any phones should be
produced to realize the maximum weekly profit?
Liliana Menjivar
Math 1520 Course Notes Sections: 6.1, 6.2