We need to find the extreme values of the function f ( x , y ) = 2 x + 4 y subject to the constraint g ( x , y ) = x 2 + y 2 − 5 = 0 . a. Show that the Lagrange equation ∇ f = λ ∇ g gives λ x = 1 and λ y = 2 b. Show that these equations imply λ ≠ 0 and y = 2 x . c. Use the constraint equation to determine the possible critical points ( x , y ) . d. Evaluate f ( x , y ) at the critical points and determine the minimum and maximum values.

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Answer 1

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Chapter 14.8, Problem 1E

To determine

We need to find the extreme values of the function $f (x, y) = 2 x + 4 y$ subject to the constraint $g (x, y) = x^{2} + y^{2} - 5 = 0$ .

a. Show that the Lagrange equation $\nabla f = λ \nabla g$ gives $λ x = 1$ and $λ y = 2$

b. Show that these equations imply $λ \neq 0$ and $y = 2 x$ .

c. Use the constraint equation to determine the possible critical points $(x, y)$ .

d. Evaluate $f (x, y)$ at the critical points and determine the minimum and maximum values.

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Answer 2

Question

Chapter 14.8, Problem 1E

To determine

We need to find the extreme values of the function $f (x, y) = 2 x + 4 y$ subject to the constraint $g (x, y) = x^{2} + y^{2} - 5 = 0$ .

a. Show that the Lagrange equation $\nabla f = λ \nabla g$ gives $λ x = 1$ and $λ y = 2$

b. Show that these equations imply $λ \neq 0$ and $y = 2 x$ .

c. Use the constraint equation to determine the possible critical points $(x, y)$ .

d. Evaluate $f (x, y)$ at the critical points and determine the minimum and maximum values.

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Answer 3

Question

Chapter 14.8, Problem 1E

To determine

We need to find the extreme values of the function $f (x, y) = 2 x + 4 y$ subject to the constraint $g (x, y) = x^{2} + y^{2} - 5 = 0$ .

a. Show that the Lagrange equation $\nabla f = λ \nabla g$ gives $λ x = 1$ and $λ y = 2$

b. Show that these equations imply $λ \neq 0$ and $y = 2 x$ .

c. Use the constraint equation to determine the possible critical points $(x, y)$ .

d. Evaluate $f (x, y)$ at the critical points and determine the minimum and maximum values.

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Answer 4

Question

Chapter 14.8, Problem 1E

To determine

We need to find the extreme values of the function $f (x, y) = 2 x + 4 y$ subject to the constraint $g (x, y) = x^{2} + y^{2} - 5 = 0$ .

a. Show that the Lagrange equation $\nabla f = λ \nabla g$ gives $λ x = 1$ and $λ y = 2$

b. Show that these equations imply $λ \neq 0$ and $y = 2 x$ .

c. Use the constraint equation to determine the possible critical points $(x, y)$ .

d. Evaluate $f (x, y)$ at the critical points and determine the minimum and maximum values.

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Answer 5

Chapter 14.8, Problem 1E

To determine

We need to find the extreme values of the function $f (x, y) = 2 x + 4 y$ subject to the constraint $g (x, y) = x^{2} + y^{2} - 5 = 0$ .

a. Show that the Lagrange equation $\nabla f = λ \nabla g$ gives $λ x = 1$ and $λ y = 2$

b. Show that these equations imply $λ \neq 0$ and $y = 2 x$ .

c. Use the constraint equation to determine the possible critical points $(x, y)$ .

d. Evaluate $f (x, y)$ at the critical points and determine the minimum and maximum values.

Answer 6

Chapter 14.8, Problem 1E

To determine

We need to find the extreme values of the function $f (x, y) = 2 x + 4 y$ subject to the constraint $g (x, y) = x^{2} + y^{2} - 5 = 0$ .

a. Show that the Lagrange equation $\nabla f = λ \nabla g$ gives $λ x = 1$ and $λ y = 2$

b. Show that these equations imply $λ \neq 0$ and $y = 2 x$ .

c. Use the constraint equation to determine the possible critical points $(x, y)$ .

d. Evaluate $f (x, y)$ at the critical points and determine the minimum and maximum values.

Answer 7

Chapter 14.8, Problem 1E

To determine

We need to find the extreme values of the function $f (x, y) = 2 x + 4 y$ subject to the constraint $g (x, y) = x^{2} + y^{2} - 5 = 0$ .

a. Show that the Lagrange equation $\nabla f = λ \nabla g$ gives $λ x = 1$ and $λ y = 2$

b. Show that these equations imply $λ \neq 0$ and $y = 2 x$ .

c. Use the constraint equation to determine the possible critical points $(x, y)$ .

d. Evaluate $f (x, y)$ at the critical points and determine the minimum and maximum values.

Answer 8

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Answer 9

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Answer 10

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Answer 11

Ch. 14.1 - Prob. 1PQ Ch. 14.1 - Prob. 2PQ Ch. 14.1 - Prob. 3PQ Ch. 14.1 - Prob. 4PQ Ch. 14.1 - Prob. 5PQ Ch. 14.1 - Prob. 1E Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E Ch. 14.1 - Prob. 5E

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Chapter 14 Solutions