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For this problem, consider (fixed, nonzero, otherwise arbitrary) real numbers a, b, c ER and the (continuous) function f(x, y) = ax² + bæy + cy?. (a) Compute Vf(x, y). (b) Show that if b² 4ac + 0, then f has a unique critical point at (x, y) = (0, 0).

For this problem, consider (fixed, nonzero, otherwise arbitrary) real numbers a, b, c ER and the (continuous) function f(x, y) = ax² + bæy + cy?. (a) Compute Vf(x, y). (b) Show that if b² 4ac + 0, then f has a unique critical point at (x, y) = (0, 0).

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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For this problem, consider (fıxed, nonzero, otherwise arbitrary) real numbers a, b, cER and the (continuous) function z = f(x,y) = ax² + bæy + cy². (a) Compute Vf(x, y). (b) Show that if b2 4ac + 0, then f has a unique critical point at (x, y) = (0, 0). Hint: It suffices to start from Vf(x, y) = (0,0) and show that (x, y) = (0,0). Be explicit about where you use the hypothesis 62 4ас — 0. - (c) Maximize z = = f(x,y) subject to the constraint g(x, y) = x² + y² = 1. Hint: The Lagrange multiplier method tells us that it suffices to solve the system of equations Vf(x, y) = AVg(x, y) and g(x, y) Hint: You should be able to show that the maximum output of f along the curve g = 1 is z= X. What is the corresponding input (x, y)?
Transcribed Image Text:For this problem, consider (fıxed, nonzero, otherwise arbitrary) real numbers a, b, cER and the (continuous) function z = f(x,y) = ax² + bæy + cy². (a) Compute Vf(x, y). (b) Show that if b2 4ac + 0, then f has a unique critical point at (x, y) = (0, 0). Hint: It suffices to start from Vf(x, y) = (0,0) and show that (x, y) = (0,0). Be explicit about where you use the hypothesis 62 4ас — 0. - (c) Maximize z = = f(x,y) subject to the constraint g(x, y) = x² + y² = 1. Hint: The Lagrange multiplier method tells us that it suffices to solve the system of equations Vf(x, y) = AVg(x, y) and g(x, y) Hint: You should be able to show that the maximum output of f along the curve g = 1 is z= X. What is the corresponding input (x, y)?
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