Goal: Use the Method of Lagrange Multipliers to find the maximum and minimum values of the function f(x, y, z) subject to the constraint that (x, y, z) lie on the sphere a2 + y? + z2 = 1. = - x + 2y – 2z We denote the constraint function as g(x, y, z) = x² + y? + z? – 1. (1) The gradients of the objective function f and the constraint function g are given by: V f(x, y, z) = ( V g(x, y, z) = ). (2) The maximum for f occurs at (x, y, z) = ( ), and the maximum value of f is (3) The minimum for f occurs at (x, y, z) = ( ), and the minimum value of f is
Goal: Use the Method of Lagrange Multipliers to find the maximum and minimum values of the function f(x, y, z) subject to the constraint that (x, y, z) lie on the sphere a2 + y? + z2 = 1. = - x + 2y – 2z We denote the constraint function as g(x, y, z) = x² + y? + z? – 1. (1) The gradients of the objective function f and the constraint function g are given by: V f(x, y, z) = ( V g(x, y, z) = ). (2) The maximum for f occurs at (x, y, z) = ( ), and the maximum value of f is (3) The minimum for f occurs at (x, y, z) = ( ), and the minimum value of f is
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:Goal: Use the Method of Lagrange Multipliers to find the maximum and minimum
values of the function
f(x, y, z)
subject to the constraint that (x, y, z) lie on the sphere
a2 + y? + z2 = 1.
= - x + 2y – 2z
We denote the constraint function as
g(x, y, z) = x² + y? + z? – 1.
(1) The gradients of the objective function f and the constraint function g are given by:
V f(x, y, z) = (
V g(x, y, z) =
).
(2) The maximum for f occurs at
(x, y, z) = (
),
and the maximum value of f is
(3) The minimum for f occurs at
(x, y, z) = (
),
and the minimum value of f is
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