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.
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.
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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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 \( O \subset \mathbb{R}^n \), \( n = 2, 3 \), and let \(\vec{r}(t)\), \( t \in I \), be a smooth parameterized curve \( C \) that lies entirely in \( O \). If \( P_0 = \vec{r}(t_0) \) is a point on \( C \) where \( f \) has a local maximum or minimum value relative to the values of \( f \) on \( C \), then:
\[
\nabla f(P_0) \cdot \frac{d\vec{r}}{dt}(t_0) = 0
\]
**Theorem 8: Lagrange Multipliers in \(\mathbb{R}^2\)**
Let \( f \) and \( g \) be differentiable functions defined on a common domain \( D \subset \mathbb{R}^2 \) which is an open set. For a given constant \( k \), let \( C \) denote the level curve defined by \( g(x, y) = k \) and assume \( \nabla g \neq 0 \) 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:
\[
f_x(P) = \lambda g_x(P), \quad f_y(P) = \lambda g_y(P), \quad g(P) = k
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Transcribed Image Text:**Theorem 6**
Let \( f \) be a differentiable function defined on an open set \( O \subset \mathbb{R}^n \), \( n = 2, 3 \), and let \(\vec{r}(t)\), \( t \in I \), be a smooth parameterized curve \( C \) that lies entirely in \( O \). If \( P_0 = \vec{r}(t_0) \) is a point on \( C \) where \( f \) has a local maximum or minimum value relative to the values of \( f \) on \( C \), then:
\[
\nabla f(P_0) \cdot \frac{d\vec{r}}{dt}(t_0) = 0
\]
**Theorem 8: Lagrange Multipliers in \(\mathbb{R}^2\)**
Let \( f \) and \( g \) be differentiable functions defined on a common domain \( D \subset \mathbb{R}^2 \) which is an open set. For a given constant \( k \), let \( C \) denote the level curve defined by \( g(x, y) = k \) and assume \( \nabla g \neq 0 \) 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:
\[
f_x(P) = \lambda g_x(P), \quad f_y(P) = \lambda g_y(P), \quad g(P) = k
\]
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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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