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Section 3.1 Extrema on an Interval
Definition: Extrema
Let f be defined on an interval I containing c.
f (c) is the minimum of f on I if f ( x) ≥ f (c) for all x in I
f (c) is the maximum of f on I if f ( x) ≤ f (c) for all x in I
The minimum and maximum of a function on an interval are the extreme values, or
extrema, of the function on the interval. The minimum and maximum of a function on an
interval are also called the absolute minimum and absolute maximum on the interval.
Theorem: 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.
Definition: Relative Extrema
If there is an open interval containing c on which f (c) is the minimum, then f (c) is
called a relative minimum of f.
If there is an open interval containing c on which f (c) is the maximum, then f (c) is
called a relative maximum of f.
Calculus I by Chinyoung Bergbauer at NHMCCD: 3.1 Extrema on an Interval
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Example 1: Find the value of the derivative at each of the relative extrema of the given
function.
a)
f ( x) = ( x − 1)( x + 2)( x − 3)
The function has 2 relative extrema, one relative maximum and one relative minimum.
The derivatives at the relative extrema are 0.
b)
f ( x) =| x − 1 |
The function has only one relative minimum at 1 and its derivative at 1 is undefined.
Calculus I by Chinyoung Bergbauer at NHMCCD: 3.1 Extrema on an Interval
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c)
f ( x) = cos x
The function has infinitely many relative extrema. The function has relative maxima at
x = 2nπ and relative minima at x = (2n + 1)π .
Definition: Critical Number
Let f be defined at c. If f ′(c) = 0 or if f is not differentiable at c, then c is a critical
number of f.
Theorem: Extrema occur only at Critical Numbers
If f has a relative maximum or minimum at c, then c is a critical number of f.
How to find (Absolute) Extrema of a continuous function on a closed interval [a,b]
1. Find the critical numbers c on the interval.
2. Evaluate all f (c) ’s, f (a) , and f (b) .
3. The least of these values is the minimum. The greatest is the maximum.
Calculus I by Chinyoung Bergbauer at NHMCCD: 3.1 Extrema on an Interval
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Example 2: Find the extrema on a closed interval.
f ( x) = 16 x 4 − 8 x 2 on [-1, 1]
Step 1: Find the critical numbers:
f ( x) = 16 x 4 − 8 x 2
f ′( x) = 64 x 3 − 16 x = 16 x(4 x 2 − 1)
= 16 x(2 x − 1)(2 x + 1) = 0 when x = 0,1 / 2,−1 / 2
f (−1) = 8
f ( 0) = 0
f (−1 / 2) = −1
f (1 / 2) = −1
f (1) = 8
The function has the absolute minimum -1 at x = ½ and -1/2.
The function has the absolute maximum 8 at x = 1 and x = -1.
Practice 2: Find the extrema on a closed interval.
f ( x) =
t
on [3, 5]
t−2
Example 3: Find the extrema of f ( x) = 4 x − 63 x 2 on [-1, 2]
f ( x ) = 4 x − 63 x 2 = 4 x − 6 x 2 / 3
f ′( x) = 4 − 4 x −1/ 3 = 4 − 3
4
43 x − 4
= 3
x
x
f ′( x) = 0 when 43 x − 4 = 0 ⇒ 4(3 x − 1) = 0 ⇒ x = 1 : critical number
f ′( x) is undefined when 3 x = 0 ⇒ x = 0 :: critical number
f ( 0) = 4( 0) − 6 3 0 2 = 0
f (1) = 4 − 63 12 = −2
f (−1) = −4 − 63 (−1) 2 = −10
f (2) = 8 − 63 2 2 = 8 − 63 4 ≈ −1.5244
The function has the maximum 0 at x = 0 and the minimum -10 at x = -1.
Example 4: Find the extrema of f ( x) = sin 2 x + 2 cos x on [0,2π ]
Calculus I by Chinyoung Bergbauer at NHMCCD: 3.1 Extrema on an Interval
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f ( x) = sin 2 x + 2 cos x
f ′( x) = 2 cos 2 x − 2 sin x = 2(1 − 2 sin 2 x) − 2 sin x
= −4 sin 2 x − 2 sin x + 2
Set f ′(x) = 0 to find the critical numbers. Note that f ′(x) is always defined.
− 4 sin 2 x − 2 sin x + 2 = 0
2 sin 2 x + sin x − 1 = 0
(2 sin x − 1)(sin x + 1) = 0
1
or sin x = −1
2
π 5π
3π
x = , ,x =
: critical numbers
2
6 6
π
π 3 3
π 
f   = sin + 2 cos =
: max
3
6
2
6
sin x =
3
3 3
5π
3
5π
 5π 
f   = sin
+ 2 cos
=−
−2
=−
: min
2
2
6
2
3
 6 
3π
 3π 
f   = sin 3π + 2 cos
=0
2
 2 
f (0) = sin 0 + 2 cos 0 = 2;
f (2π ) = sin 4π + 2 cos 2π = 2
Practice 3: Find the extrema of f ( x) = 33 x 2 − 2 x on [−1,1]
Practice 4: Find the extrema of f ( x) = x 2 − 2 − cos x on [−1,3]
Calculus I by Chinyoung Bergbauer at NHMCCD: 3.1 Extrema on an Interval
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Calculus I by Chinyoung Bergbauer at NHMCCD: 3.1 Extrema on an Interval
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