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 ID: 25921 Download Pdf

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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

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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 ( ). Today’s goal: Given a function , identify its local maxima and minima. Math 105 (Section 203) Multivariable Calculus – Extremization 2010W T2 1 / 6

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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 ( ). Today’s goal: Given a function , identify its local maxima and minima. Math 105 (Section 203) Multivariable Calculus – Extremization 2010W T2 1 / 6

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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

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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

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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

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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

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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

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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 company’s proﬁts. A. (100 80) B. (120 90) C. (120 80) D. (80 120) Math 105 (Section 203) Multivariable Calculus – Extremization 2010W T2 6 / 6

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