MATH 100 Lecture 3: Math 100 notes - Extrema on an Interval
21 May 2018
School
Department
Course
Professor
Math 100
Lesson 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 an interval I when f(c) f(x) for all x on I
2. f(x) is the maximum of f on an interval I when f(c) f(x) for all x on 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 absolute min. and absolute max. or the global min. and global
max. on the interval. Extrema can occur at interior points or endpoints of an interval.
Endpoint extrema are ones that happen at the endpoints.
• Theorem 3.1: Extreme Value Theorem
If f is continuous on a closed interval [a,b] , then f has both a min. and a max. on the
interval.
• Critical numbers and definition of relative extrema:
o 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 extrema (maximum)
2. If there is an open interval containing C on which f(c) is a minimum, then f(c) is
called a relative extrema (minimum)
*Note: Relative extrema are sometimes called local extrema.
o Definition of critical numbers:
Let f e defined at C. If f’ = 0 or if it is not differentiale at C, then C is a critical
number.
• Theorem 3.2: Relative Extrema Occur Only at Critical Numbers
If f has a relative maximum or minimum at x = C, C is a critical number of f.
o Guidelines for finding extrema on a closed interval:
1. Find 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 these is the minimum, and the greatest is the maximum
find more resources at oneclass.com
find more resources at oneclass.com
Document Summary
Lesson 3. 1 (cid:862)extrema on an interval(cid:863: definition of extrema: 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 absolute min. and absolute max. or the global min. and global max. on the interval. Extrema can occur at interior points or endpoints of an interval. Endpoint extrema are ones that happen at the endpoints: theorem 3. 1: extreme value theorem. *note: relative extrema are sometimes called local extrema: definition of critical numbers: Let f (cid:271)e defined at c. if f"(cid:894)(cid:272)(cid:895) = 0 or if it is not differentia(cid:271)le at c, then c is a critical number: theorem 3. 2: relative extrema occur only at critical numbers. Example 1: find the extrema of (cid:4666)(cid:4667)=(cid:885)(cid:2872) (cid:886)(cid:2871) on [-1,2]. Left endpoint f(-1) = 7 f(0) = 0 f(1) = -1 (minimum) Both answers are within the closed interval [-1,2].
