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(b) Write down the Lagrangian for the problem of finding the critical points of the function f of part (a), subject to the constraint h = 2. Write down the first order conditions, and find all 8 stationary points for the function f(x, y) subject to the given constraint.

(b) Write down the Lagrangian for the problem of finding the critical points of the function f of part (a), subject to the constraint h = 2. Write down the first order conditions, and find all 8 stationary points for the function f(x, y) subject to the given constraint.

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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can you please do part b , can you find all 8 coordiantes and show teh working please

2. (a) Consider the function f(x, y) = xy − 3x²y² = with x, y constrained by the relation h(x, y) x² + y² 2. Find the partial derivatives fæ, fy and the total derivatives df/dx and df/dy, and explain the difference between partial and total derivatives geometrically (if possible through a sketch). Answer: The partial derivatives of f are given by = fx=y - 6xy², fy = x - 6x²y, and from he constraint h = 2 by implicit differentiation we get dy/dx equivalently dx/dy=-y/x. Thus, the total derivatives are = -x/y, or df df = y − 6xy² — (x − 6x²y) ², = x − 6x²y — (y — 6xy²) 4/ dx Y dy (b) Write down the Lagrangian for the problem of finding the critical points of the function f of part (a), subject to the constraint h = 2. Write down the first order conditions, and find all 8 stationary points for the function f(x, y) subject to the given constraint. Hint: It may be useful to consider sums and differences of the first order conditions on the Lagrangian.
Transcribed Image Text:2. (a) Consider the function f(x, y) = xy − 3x²y² = with x, y constrained by the relation h(x, y) x² + y² 2. Find the partial derivatives fæ, fy and the total derivatives df/dx and df/dy, and explain the difference between partial and total derivatives geometrically (if possible through a sketch). Answer: The partial derivatives of f are given by = fx=y - 6xy², fy = x - 6x²y, and from he constraint h = 2 by implicit differentiation we get dy/dx equivalently dx/dy=-y/x. Thus, the total derivatives are = -x/y, or df df = y − 6xy² — (x − 6x²y) ², = x − 6x²y — (y — 6xy²) 4/ dx Y dy (b) Write down the Lagrangian for the problem of finding the critical points of the function f of part (a), subject to the constraint h = 2. Write down the first order conditions, and find all 8 stationary points for the function f(x, y) subject to the given constraint. Hint: It may be useful to consider sums and differences of the first order conditions on the Lagrangian.
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