Relative maxima and minina Denition Given a function  of two variables We say that has a local maximum at the point   if for all   close enough to
88K - views

Relative maxima and minina Denition Given a function of two variables We say that has a local maximum at the point if for all close enough to

We say that has a local minimum at the point if for all close enough to Todays goal Given a function identify its local maxima and minima Math 105 Section 203 Multivariable Calculus Extremization 2010W T2 1 6 brPage 2br Relative maxima and

Tags : say that has
Download Pdf

Relative maxima and minina Denition Given a function of two variables We say that has a local maximum at the point if for all close enough to




Download Pdf - The PPT/PDF document "Relative maxima and minina Denition Give..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.



Presentation on theme: "Relative maxima and minina Denition Given a function of two variables We say that has a local maximum at the point if for all close enough to"— Presentation transcript:


Page 1
Relative maxima and minina Definition Given a function ) of two variables, We say that has a local maximum at the point ( ) if for all ( ) close enough to ( ). We say that has a local minimum at the point ( ) if for all ( ) close enough to ( ). Todays goal: Given a function , identify its local maxima and minima. Math 105 (Section 203) Multivariable Calculus Extremization 2010W T2 1 / 6
Page 2
Relative maxima and minina Definition Given a function ) of two variables, We say that has a local maximum at the point ( ) if for all ( ) close enough to ( ).

We say that has a local minimum at the point ( ) if for all ( ) close enough to ( ). Todays goal: Given a function , identify its local maxima and minima. Math 105 (Section 203) Multivariable Calculus Extremization 2010W T2 1 / 6
Page 3
The first derivative test Description of the test The first step in finding local max or min of a function ) is to find points ( ) that satisfy the two equations ) = 0 and ) = 0 Any such point ( ) is called a critical point of Note: Any local max or min of has to be a critical point, but every critical point need not be a

local max or min. Math 105 (Section 203) Multivariable Calculus Extremization 2010W T2 2 / 6
Page 4
The first derivative test Description of the test The first step in finding local max or min of a function ) is to find points ( ) that satisfy the two equations ) = 0 and ) = 0 Any such point ( ) is called a critical point of Note: Any local max or min of has to be a critical point, but every critical point need not be a local max or min. Math 105 (Section 203) Multivariable Calculus Extremization 2010W T2 2 / 6
Page 5
Finding critical points : an

example Find all critical points of the following function ) = xy A. (0 0) (1 1) B. (1 1) C. (0 0) (1 1) (1 1) 1) 1) D. There is no critical point E. (0 0) Math 105 (Section 203) Multivariable Calculus Extremization 2010W T2 3 / 6
Page 6
The previous example (ctd) Is the critical point (1 1) a local max, a local min or neither? The second derivative test If ( ) is a critical point of , calculate ), where xx yy xy 1. If 0 and xx 0, then has a local maximum value at ). 2. If 0 and xx 0, then has a local minimum value at ). 3. If 0, then has a saddle point at ( ). 4. If ) = 0, then

the test is inconclusive. Math 105 (Section 203) Multivariable Calculus Extremization 2010W T2 4 / 6
Page 7
Classifying critical points : an example In the example ) = xy determine whether the critical point (1 1) is A. a local minimum B. a local maximum C. a saddle point D. neither of the above Math 105 (Section 203) Multivariable Calculus Extremization 2010W T2 5 / 6
Page 8
An application A company manufactures two products and that sell for $10 and $9 per unit respectively. The cost of producing units of and units of is 400 + 2 + 3 + 0 01(3 xy + 3 Find the values of

and that maximize the companys profits. A. (100 80) B. (120 90) C. (120 80) D. (80 120) Math 105 (Section 203) Multivariable Calculus Extremization 2010W T2 6 / 6