3.1 Extrema on an Interval
The textbook gives the following example at the start of chapter 4:
The mileage of a certain car can be approximated by:
m  v   0.00015v3  0.032v2  1.8v  1.7
At what speed should you drive the car to obtain the best gas
mileage?
Of course, this problem isn’t entirely realistic, since it is
unlikely that you would have an equation like this for your
car.
We could solve the problem graphically:
m  v   0.00015v3  0.032v2  1.8v  1.7
We could solve the problem graphically:
m  v   0.00015v3  0.032v2  1.8v  1.7
On the TI-89, we use F5 (math), 4: Maximum, choose lower and upper bounds,
and the calculator finds our answer.
We could solve the problem graphically:
m  v   0.00015v3  0.032v2  1.8v  1.7
On the TI-89, we use F5 (math), 4: Maximum, choose lower and upper bounds,
The car will get approximately 32 miles per gallon when driven at 38.6
and the calculator finds our answer.
miles per hour.
m  v   0.00015v3  0.032v2  1.8v  1.7
Notice that at the top
of the curve, the
horizontal tangent has
a slope of zero.
Traditionally, this fact has been used both as an aid to graphing by hand and as a
method to find maximum (and minimum) values of functions.
Even though the graphing calculator and the computer have eliminated the
need to routinely use calculus to graph by hand and to find maximum and
minimum values of functions, we still study the methods to increase our
understanding of functions and the mathematics involved.
Absolute extreme values are either maximum or minimum points on a
curve.
They are sometimes called global extremes.
They are also sometimes called absolute extrema.
(Extrema is the plural of the Latin extremum.)
Absolute Maximum and Minimum
Definition:
• A function f has an absolute maximum (or global
maximum) at c if f(c) ≥ f(x) for all x in D, where D is the
domain of f. The number f(c) is called maximum value
of f on D.
• Similarly, f has an absolute minimum at c if f(c) ≤ f(x)
for all x in D and the number f(c) is called the minimum
value of f on D.
• The maximum and minimum values of f are called the
extreme values of f.
Local Maximum and Minimum
Definition:
• A function f has a local maximum (or relative
maximum) at c if f(c) ≥ f(x) when x is near c.
• Similarly, f has a local minimum at c if f(c) ≤ f(x)
when x is near c.
Example:
Absolute maximum
(also local maximum)
Local maximum
Local minimum
Absolute minimum
(also local minimum)
Local minimum
Extreme Value Theorem:
If f is continuous over a closed interval, then f has
absolute maximum and minimum over that interval.
Maximum &
minimum
at interior points
Maximum &
minimum
at endpoints
Maximum at
interior point,
minimum at
endpoint
Example when a continuous function doesn’t have minimum
or maximum because the interval is not closed.
4
No
Maximum
3
2
1
yx
2
D   0, 2 
-2
-1
0
1
2
No Minimum
Suppose we know that extreme values exist. How to find them?
Absolute maximum
(also local maximum)
Local maximum
Local minimum
Notice that local extremes in the interior of the function occur
where f  is zero or f  is undefined.
Fermat’s Theorem
If f has a local maximum or minimum at c,
and if f ′(c) exists, then f ′(c) = 0 .
Note: When f ′(c) = 0 , f doesn’t necessarily have a
maximum or minimum at c. (In other words, the
converse of Fermat’s Theorem is false in general).
Example when f ′(c) = 0 but f has no maximum or minimum at c .
2
yx
3
1
-2
-1
0
1
2
f0
-1
-2
(not an extreme)
Critical numbers
• Definition: A critical number of a function f is a number c
in the domain of f such that either f ′(c) = 0 or f ′(c)
doesn’t exist.
• Example: Find the critical numbers of f(x) = x1/2(x-3)
Solution:
x 3
x  3  2 x 3x  3
f ( x) 
2 x
 x
2 x

2 x
Thus, the critical numbers are 0 and 1.
• If f has a local maximum or minimum at c , then c is a
critical number of f .
The closed interval method for finding
absolute maximum or minimum
To find the absolute maximum and minimum values
of a continuous function f on a closed interval
[a,b]:
1. Find the values of f at the critical numbers of f in
(a,b) .
2. Find the values of f at the endpoints of the
interval.
3. The largest of the values from Steps 1 and 2 is
the absolute maximum value; the smallest of
these values is the absolute minimum value.
Examples
2x + 5
Find the extrema of f ( x) =
from [ 0,5 ]
3
2x + 5
f ( x) =
3
2
f ¢( x) = ® no critical numbers, so we have to look at just the endpoints
3
5
æ 5æ
f ( 0 ) = , f ( 5 ) = 5 ® æ0, æ is a minimum and ( 5,5 ) is a maximum
æ 3æ
3
f ( x) = x2 + 2x - 4, [ -1,1]
f ¢( x) = 2x + 2 ® 0 = 2x + 2 ® x = -1 is a critical number
f ( -1) = -5
f (1) = -1
Therefore ( -1,-5 ) is a minimum and (1,-1) is a maximum.
f ( x) = x3 -12x, [ 0, 4 ]
f ¢( x) = 3x2 -12 = 3( x2 - 4 ) ® Critical points are -2 and 2, but -2 is
not in the interval.
f ( 0) = 0
f ( 4 ) = 16 ® maximum is at ( 4,16 )
f ( 2 ) = -16 ® minimum is at ( 2,-16 )
HW Pg. 169 1-6, 23-39 odds, 55-59