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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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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