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202021: Lagrange multipliers. a. Optimise f(x, y, z) = xy + yz + zx subject to x + y + z = A for the following 1.01. values of A: A = 1 and A: =

202021: Lagrange multipliers. a. Optimise f(x, y, z) = xy + yz + zx subject to x + y + z = A for the following 1.01. values of A: A = 1 and A: =

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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202021: Lagrange multipliers. a. Optimise f(x, y, z) = xy + yz + zx subject to x + y + z = A for the following values of A: A = 1 and A = 1.01. : b. Let f* RR be defined as f*(A) is the optimal value of f as found in part a for a given value of A. Using your answers from part a, compute f*(1.01) – f*(1) 0.01 c. Investigate at least two further cases of values of A and A+ 0.01, computing the ratio f*(A + 0.01) — ƒ* (A) 0.01 You need not show full working for this investigation, but summarise your results in a table structured as follows. (Recall that is the Lagrange multiplier.) Record all your answers to at least two decimal places. A X f* (A) 1 f* (A+0.01)-f* (A) 0.01 Look carefully over your workings and your results: do you spot any pattern? d. Solve the general case of the optimisation problem in part a, where the con- straint is x + y + z = A, and hence explain why the pattern holds. e. Generalise your result from the above investigation (parts b to d), applicable to all Lagrange multiplier problems with an objective function f : R" → R and a single constraint function, g: R" → R, with g(x) = A. f. Prove your generalisation from part e above.
Transcribed Image Text:202021: Lagrange multipliers. a. Optimise f(x, y, z) = xy + yz + zx subject to x + y + z = A for the following values of A: A = 1 and A = 1.01. : b. Let f* RR be defined as f*(A) is the optimal value of f as found in part a for a given value of A. Using your answers from part a, compute f*(1.01) – f*(1) 0.01 c. Investigate at least two further cases of values of A and A+ 0.01, computing the ratio f*(A + 0.01) — ƒ* (A) 0.01 You need not show full working for this investigation, but summarise your results in a table structured as follows. (Recall that is the Lagrange multiplier.) Record all your answers to at least two decimal places. A X f* (A) 1 f* (A+0.01)-f* (A) 0.01 Look carefully over your workings and your results: do you spot any pattern? d. Solve the general case of the optimisation problem in part a, where the con- straint is x + y + z = A, and hence explain why the pattern holds. e. Generalise your result from the above investigation (parts b to d), applicable to all Lagrange multiplier problems with an objective function f : R" → R and a single constraint function, g: R" → R, with g(x) = A. f. Prove your generalisation from part e above.
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