Explanation Given: The given functions are f ( x , y , z ) = x 2 + y 2 + z 2 and g ( x , y , z ) = 2 x + 3 y − 5 z + 4 = 0 . Definition used: “The Lagrange multipliers defined as ∇ f ( x , y , z ) = λ ∇ g ( x , y , z ) . This equation can be expressed as f x = λ g x , f y = λ g y , f z = λ g z and g ( x , y , z ) = k ”. Calculation: Themaximum or minimum value of the function f ( x , y , z ) = x 2 + y 2 + z 2 subject to the constraint g ( x , y , z ) = 2 x + 3 y − 5 z + 4 = 0 . The Lagrange multipliers ∇ f ( x , y , z ) = λ ∇ g ( x , y , z ) is computed as follows. ∇ f ( x , y , z ) = λ ∇ g ( x , y , z ) 〈 f x , f y , f z 〉 = λ 〈 g x , g y , g z 〉 〈

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ISBN: 9781323178522

Author: Briggs

Publisher: PEARSON

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Chapter 12.9, Problem 3E

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To write: The Lagrange multiplier conditions that must be satisfied by a point when it maximizes or minimizes the function f subject to g(x,y,z)=0 .

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Chapter 12.9, Problem 2E

Chapter 12.9, Problem 4E

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4) A company is trying to determine the optimal sale price, p, for its new widget to maximize profits (in USD). It is known that the total sales can be estimated by S(p) = 51000 p - 150 p² (USD), and the total costs can be estimated by C(p) = 6,500,000 -19,000 P (USD). a) If the profit is G(p) = S(p) - C(p), substitute the above equations to determine the profit G(p). Express your result for G(p) in the standard form for a quadratic equation. b) Determine the sales price(s) p that would result in zero profit G(p) = 0 by both completing the and the quadratic formula. square c) Using your results from part b) determine the price that would result in maximum profit and calculate the corresponding maximum profit. c) Use your results from parts a) to c) to plot the total profits G(p) in Excel and paste a copy into Word. Be sure to properly label the graph and clearly indicate the price corresponding to the maximum profit on your graph.

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The Lagrange multiplier conditions that must be satisfied by a point when it maximizes or minimizes the function f subject to g ( x , y , z ) = 0 .

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To write: The Lagrange multiplier conditions that must be satisfied by a point when it maximizes or minimizes the function f subject to g(x,y,z)=0 .

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Chapter 12 Solutions