Suppose that a temperature of a metal plate is given by T(x,y)=x2+2x+y2, for points (x,y) on the elliptic plate defined by 6x2+5y2≤60. Find the maximum and minimum temperatures on the plate. Use Lagrange Multipliers to determine the absolute extrema of f on the indicated constraint. Theorem 6.Let f be a differentiable function defined on an open set O c R", n = 2,3 and let 7 (t), tel,be a smooth parameterized curve C that lies entirely in O. If Po= F(to) is a point on C wheref has a local maximum or minimum value relative to the values off on C, thenof (P₁)(t-(to) = 0(4)Theorem 8. Lagrange Multipliers in R²Let f and g be differentiable functions defined on a common domain Dc R² which is anopen set. For a given constant k let C denote the level curve defined by g(x, y) = k and assumeVg 0 on C. If there is a point P on C where f restricted to the level curve C has a localmaximum or minimum value, then P satisfies the following three equationsfx(P) = Agx(P), fy(P) = Ag,(P), g(P) = k

The temperature of a metal plate is given by T(x,y)=x2+2x+y2 for points (x,y) on the elliptic plate…

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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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Question

Suppose that a temperature of a metal plate is given by T(x,y)=x²+2x+y², for points (x,y) on the
elliptic plate defined by 6x²+5y²≤60. Find the maximum and minimum temperatures on the plate.
Use Lagrange Multipliers to determine the absolute extrema of f on the indicated constraint.

Theorem 6.
Let f be a differentiable function defined on an open set O c R", n = 2,3 and let 7 (t), tel,
be a smooth parameterized curve C that lies entirely in O. If Po= F(to) is a point on C where
f has a local maximum or minimum value relative to the values off on C, then
of (P₁)
(t
-(to) = 0
(4)
Theorem 8. Lagrange Multipliers in R²
Let f and g be differentiable functions defined on a common domain Dc R² which is an
open set. For a given constant k let C denote the level curve defined by g(x, y) = k and assume
Vg 0 on C. If there is a point P on C where f restricted to the level curve C has a local
maximum or minimum value, then P satisfies the following three equations
fx(P) = Agx(P), fy(P) = Ag,(P), g(P) = k

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Are there any more local or global maximums or minimums?

Use Lagrange Multipliers to determine the absolute extrema of f on the indicated constraint and label which ones are absolute extrema (global maximums, and global minimums).

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