i) Consider maximising the function f(x, y) = 2x + 3y subject to the simultaneous constraints x ≤y+1, 2x ≥y-3, and y≤ 3. Write down the relevant Lagrangian and the corresponding first order equations, including the complementary slackness conditions, and the relevant inequalities. Analyse these to find the unique point that gives the maximiser, and the corresponding values of the Lagrange multipliers (which constraints are binding?). Sketch a graph indicating the region defined by the inequalities, and explain the solution found in terms of the level curves (i.e., level lines) of f. What is the maximum value of f under the constraints?
i) Consider maximising the function f(x, y) = 2x + 3y subject to the simultaneous constraints x ≤y+1, 2x ≥y-3, and y≤ 3. Write down the relevant Lagrangian and the corresponding first order equations, including the complementary slackness conditions, and the relevant inequalities. Analyse these to find the unique point that gives the maximiser, and the corresponding values of the Lagrange multipliers (which constraints are binding?). Sketch a graph indicating the region defined by the inequalities, and explain the solution found in terms of the level curves (i.e., level lines) of f. What is the maximum value of f under the constraints?
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:A1.
i) Consider maximising the function f(x, y) = 2x + 3y subject to the simultaneous constraints
x≤y+1, 2x ≥y-3, and y≤ 3.
Write down the relevant Lagrangian and the corresponding first order equations, including
the complementary slackness conditions, and the relevant inequalities. Analyse these to find
the unique point that gives the maximiser, and the corresponding values of the Lagrange
multipliers (which constraints are binding?). Sketch a graph indicating the region defined by
the inequalities, and explain the solution found in terms of the level curves (i.e., level lines)
of f. What is the maximum value of ƒ under the constraints?
ii) Find all seven solution candidates for maximising
subject to
ƒ(x, y, z) = x² + 4y² + 9z²
g(x, y, z)
=
x² + y² + z² ≤ 1
by forming the relevant Lagrangian and applying the first order conditions and the comple-
mentary slackness condition. For which solution candidates is the constraint binding, and
what is the maximum?
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