The paraboloid z = x² + y² intersects the plane x + y + z = 1 in the shape of an ellipse. The objective of this problem is to find the point on the ellipse closest and furtherest away from the origin. To this end, we establish the following constrained optimization problem: max./min. f(x, y, z) = √√√x² + y² + z² z = x² + y² x+y+z=1 • New objective: g(x, y) • New constraint: s.t. To reduce the problem from three variables to two (i.e., removing z), substitute the first constraint into the objective function and square the result, then substitute the first constraint into the second constraint.
The paraboloid z = x² + y² intersects the plane x + y + z = 1 in the shape of an ellipse. The objective of this problem is to find the point on the ellipse closest and furtherest away from the origin. To this end, we establish the following constrained optimization problem: max./min. f(x, y, z) = √√√x² + y² + z² z = x² + y² x+y+z=1 • New objective: g(x, y) • New constraint: s.t. To reduce the problem from three variables to two (i.e., removing z), substitute the first constraint into the objective function and square the result, then substitute the first constraint into the second constraint.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Transcribed Image Text:The paraboloid z = x² + y² intersects the plane x + y + z = 1 in the shape of an ellipse.
The objective of this problem is to find the point on the ellipse closest and furtherest away from the origin.
To this end, we establish the following constrained optimization problem:
• New objective: g(x, y)
• New constraint:
max./min. f(x, y, z) = √√x² + y² + z²
z = x² + y²
x+y+z=1
To reduce the problem from three variables to two (i.e., removing z), substitute the first constraint into
the objective function and square the result, then substitute the first constraint into the second constraint.
=
s.t.
We have now constructed an equivalent optimization problem.
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