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Two parabolas are given: - C₁ is the curve of intersection between y = 1-2² and 2 = - C₂ is the curve of intersection between z=1-2² and y = 1 + z. Find the points on C₁ and C₂ which are closest to one another.

Two parabolas are given: - C₁ is the curve of intersection between y = 1-2² and 2 = - C₂ is the curve of intersection between z=1-2² and y = 1 + z. Find the points on C₁ and C₂ which are closest to one another.

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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I need help with the following question about Optimization (Vector Calculus)
Question 3. Optimization: (a) (Parabolic Distancing) Two parabolas are given: - C₁ is the curve of intersection between y = 1-² and z = ². - C₂ is the curve of intersection between x = 1 - 2² and y = 1 + z. Find the points on C₁ and C₂ which are closest to one another. (b) (Optimizing under monotone composition) (i) Suppose that H: R→ R is a real-valued and strictly increasing function Consider h(x, y) = H (x² + y²), and let KC R² be any subset of the plane. Prove that (zo, yo) E K satisfies that h(zo, yo) ≥ h(x, y) for all (x, y) € K if and only if x + y 2x² + y² for all (x, y) = K (ii) Find the location and value of the absolute maximum of h(x, y) - [²2² +0³ = 70 -t² dt among points (x, y) = R² satisfying 2x² + 3xy=1-2y². You can leave your final answer for the absolute maximum value in the form of an integral.
Transcribed Image Text:Question 3. Optimization: (a) (Parabolic Distancing) Two parabolas are given: - C₁ is the curve of intersection between y = 1-² and z = ². - C₂ is the curve of intersection between x = 1 - 2² and y = 1 + z. Find the points on C₁ and C₂ which are closest to one another. (b) (Optimizing under monotone composition) (i) Suppose that H: R→ R is a real-valued and strictly increasing function Consider h(x, y) = H (x² + y²), and let KC R² be any subset of the plane. Prove that (zo, yo) E K satisfies that h(zo, yo) ≥ h(x, y) for all (x, y) € K if and only if x + y 2x² + y² for all (x, y) = K (ii) Find the location and value of the absolute maximum of h(x, y) - [²2² +0³ = 70 -t² dt among points (x, y) = R² satisfying 2x² + 3xy=1-2y². You can leave your final answer for the absolute maximum value in the form of an integral.
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