Use Lagrange multipliers to prove that the rectangle with maximum area that has a given perimeter p is a square. Let the sides of the rectangle be x and y and let f and g represent the area (A) and perimeter (p), respectively. Find the following. A = f(x, y) = xy p = g(x, y) = 2(x+y) V(x, y) =yi+xi Vg = yi +xi Then i = implies that x =y 2 Therefore, the rectangle with maximum area is a square with side length equal
Use Lagrange multipliers to prove that the rectangle with maximum area that has a given perimeter p is a square. Let the sides of the rectangle be x and y and let f and g represent the area (A) and perimeter (p), respectively. Find the following. A = f(x, y) = xy p = g(x, y) = 2(x+y) V(x, y) =yi+xi Vg = yi +xi Then i = implies that x =y 2 Therefore, the rectangle with maximum area is a square with side length equal
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:Use Lagrange multipliers to prove that the rectangle with maximum area that has a given perimeter p is a square.
Let the sides of the rectangle be x and y and let f and g represent the area (A) and perimeter (p), respectively. Find the following.
A = f(x, y) = xy
p = g(x, y) =
2 (x +y)
V(x, y) =yi+xi
Vg = yi +xi
Then i =
implies that x =y
2
Therefore, the rectangle with maximum area is a square with side length equai
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