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 f ( x ) ≥ f ( c 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 called a relative maximum of f . f ( c ) is the maximum, then f ( c ) is
Calculus I by Chinyoung Bergbauer at NHMCCD: 3.1 Extrema on an Interval 1

Example 1: Find the value of the derivative at each of the relative extrema of the given function.
The function has 2 relative extrema, one relative maximum and one relative minimum.
The derivatives at the relative extrema are 0.
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 2

c) f ( x ) = cos x
The function has infinitely many relative extrema. The function has relative maxima at x = 2 n π and relative minima at x = ( 2 n + 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 3

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 − t
2
on [3, 5]
Example 3: Find the extrema of f ( x ) = 4 x − 6 3 x 2 = 4 x − 6 x 2 / 3 f ( x ) = 4 x − 6 3 x 2 on [-1, 2] f ′ ( x ) = 4 − 4 x − 1 / 3 f ′ ( x ) = 0 when 4 3
= 4 −
3
4 x x − 4 = 0
=
4 3
3 x
⇒ 4 ( 3 x
− 4 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 − 6 3 1 2 = − 2 f ( − 1 ) = − 4 − 6 3 ( − 1 ) 2 = − 10 f ( 2 ) = 8 − 6 3 2 2 = 8 − 6 3 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 , 2 π ]
Calculus I by Chinyoung Bergbauer at NHMCCD: 3.1 Extrema on an Interval 4

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 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 sin x x =
π
6
,
=
1 or sin
2
5 π
6
, x = x
3 π
2
= − 1
: critical numbers f
π
6
= sin
π
3
+ 2 cos
π
6
=
3
2
3
: max f f
(
5 π
6
2 π )
= sin
5 π
= sin 4 π
3
+ 2 cos
5 π
6 f
3 π
= sin 3 π + 2 cos
3 π
2 f ( 0 ) = sin 0 + 2 cos 0 = 2 ;
2
+ 2 cos 2 π
= −
= 0
= 2
2
3
− 2
2
3
= −
3
2
3
: min
Practice 3: Find the extrema of
Practice 4: Find the extrema of f f ( x ) = 3 3 x 2
( x ) = x 2 − 2
− 2 x on [ − 1 , 1 ]
− cos x on [ − 1 , 3 ]
Calculus I by Chinyoung Bergbauer at NHMCCD: 3.1 Extrema on an Interval 5

Calculus I by Chinyoung Bergbauer at NHMCCD: 3.1 Extrema on an Interval 6