Section 3.1
Extrema on an Interval
Definition of Extrema
Let f be defined on an interval I containing c.
1. f (c) is the minimum of f on I if
f (c) ≤ f (x) for all x in I.
2. f (c) is the maximum of f on I if
f (c) ≥ f (x) for all x in I.
The minimum and maximum of a function on an
interval are the extreme values or extrema (the
singular form of extrema is extremum), of the
function on the interval.
The minimum and maximum of a function on an
interval are also called the absolute (global)
minimum and absolute (global) maximum on the
interval.
Extrema can occur at interior points or endpoints of
an interval. Extrema that occur at the endpoints are
endpoint extrema. End point extrema can only be
absolute extrema.
Definition of Relative Extrema
1.
If there is an open interval containing c on
which f (c) is a maximum, then f (c) is
called a relative (local) maximum of f, or
you can say that f has a relative (local)
maximum at (c , f (c)).
2.
If there is an open interval containing c on
which f (c) is a minimum, then f (c) is called a
relative (local) minimum of f , or you can
say that f has a relative (local) minimum at
(c, f (c)).
The plural of relative maximum is relative maxima,
and the plural of relative minimum is relative
minima.
If a relative minimum (valley) or a relative
maximum (hill) is smooth and rounded, then the
graph has a horizontal tangent line.
If a relative minimum (valley) or a relative
maximum (hill) is sharp and peaked, then the graph
represents a function that is not differentiable at
the high point (low point).
The Extreme Value Theorem
If f is continuous on a closed interval [a, b], then
f has both a minimum and a maximum on the
interval.
Example 1
Remember relative (local) minimum or
maximum can never be at the endpoints on a
closed interval.
(12, 7)
(7, 5)
(−2, 2)
(9, 4)
(−4, −1)
(3, −3)
The graph shows a closed interval [−4, 12].
Find the relative (local) minima and maxima. Find
the absolute(global) minimum and maximum, if
possible.
(12, 7)
(7, 5)
(−2, 2)
(9, 4)
(−4, −1)
(3, −3)
The local minima are:
(3, −3) and (9, 4)
(12, 7)
(7, 5)
(−2, 2)
(9, 4)
(−4, −1)
(3, −3)
The local maxima are:
(−2, 2) and (7, 5)
(12, 7)
(7, 5)
(−2, 2)
(9, 4)
(−4, −1)
(3, −3)
The absolute minimum is:
(3, −3)
(12, 7)
(7, 5)
(−2, 2)
(9, 4)
(−4, −1)
(3, −3)
The absolute maximum is:
(12, 7)
Definition of a Critical Number
Let f be defined at c. If f ꞌ(c) = 0 or if f is not
differentiable at c, then c is a critical number of
f.
Relative Extrema Occur Only at
Critical Numbers
If f has a relative minimum or relative maximum at
x = c, then c is a critical number.
Guidelines for Finding Extrema on
a Closed Interval
To find the extrema of a continuous function f on a
closed interval [a, b], use the following steps.
1. Find the critical numbers of f in (a, b).
2. Evaluate f at each critical number in (a, b).
3. Evaluate f at each endpoint of [a, b].
4. The least of the values is the minimum. The
greatest is the maximum.
Example 2
Find the extrema of f (x) = 3x4 – 4x3 on the
interval [−1, 2].
This is also called the candidates test.
1.
2.
Find the critical numbers of f (x) = 3x4 – 4x3 in
(−1, 2).
f ꞌ(x) = 12x3 – 12x2
12x3 – 12x2 = 0
12x2(x – 1) = 0
x = 0, 1
Evaluate f (0) and f (1).
f (0) = 3(0)4 – 4(0)3 = 0
f (1) = 3(1)4 – 4(1)3 = −1
3.
4.
Evaluate f (−1) and f (2).
f (−1) = 3(−1)4 – 4(−1)3 = 7
f (2) = 3(2)4 – 4(2)3 = 16
The absolute minimum is f (1) = −1.
The absolute maximum is f (2) = 16.
Example 3
Find the extrema of the function
h(x) = sin2x + cos x, [0, 2π]
Step 1: Find the critical #s in (0, 2π).
h '  x   2sin x cos x  sin x
2sin x cos x  sin x  0
sin x  2cos x  1  0
sin x  0
x
2cos x  1  0
1
cos x 
2
 5
x ,
3 3
Step 2: Evaluate f at each critical number in (0, 2π).
h     sin 2     cos   
 0 1
 1


2
h    sin    cos  
3
3
3
2
 3 1

 
 2  2
5

4
 5  5
h  
 3  4
Step 3: Evaluate f at each endpoint of [0, 2π].
f 0  1
f  2   1
Step 4: The least of these values is the minimum.
The greatest is the maximum.
The absolute minimum value is f (π) = −1 and the
absolute maximum value is

f  
3
 5 
f    1.25.
 3 