The point P ( x , y ) that minimize the sum of distances between the fixed points lying in a plane.

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Answer 1

Question

Chapter 14.7, Problem 64E

To determine

The point P(x, y) that minimize the sum of distances between the fixed points lying in a plane.

To determine

(a)

To Show:

That the condition $\nabla f (P) = 0$ is equivalent to $e + f + g = 0$ .

To determine

(b)

To Show:

That the function $f (x, y)$ is differentiable except at points A, B and C and the minimum of $f (x, y)$ occurs either at P satisfying the equation $e + f + g = 0$ , where e, f and g are the unit vectors in the direction of vectors $\vec{P A}, \vec{P B}$ and $\vec{P C}$ .

To determine

(c)

To Prove:

That the equation $e + f + g = 0$ , where e, f and g are the unit vectors in the direction of vectors holds if and only if P is the Fermat point, defined as the point P for which the angles between the segments $\bar{A P}, \bar{B P}$ and $\bar{C P}$ are all 120°.

To determine

(d)

To Show:

That the Fermat point does not exist if one of the angles in $Δ A B C$ is greater than 120° and the point which exist in this case is different from Fermat point.

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Answer 2

Question

Chapter 14.7, Problem 64E

To determine

The point P(x, y) that minimize the sum of distances between the fixed points lying in a plane.

To determine

(a)

To Show:

That the condition $\nabla f (P) = 0$ is equivalent to $e + f + g = 0$ .

To determine

(b)

To Show:

That the function $f (x, y)$ is differentiable except at points A, B and C and the minimum of $f (x, y)$ occurs either at P satisfying the equation $e + f + g = 0$ , where e, f and g are the unit vectors in the direction of vectors $\vec{P A}, \vec{P B}$ and $\vec{P C}$ .

To determine

(c)

To Prove:

That the equation $e + f + g = 0$ , where e, f and g are the unit vectors in the direction of vectors holds if and only if P is the Fermat point, defined as the point P for which the angles between the segments $\bar{A P}, \bar{B P}$ and $\bar{C P}$ are all 120°.

To determine

(d)

To Show:

That the Fermat point does not exist if one of the angles in $Δ A B C$ is greater than 120° and the point which exist in this case is different from Fermat point.

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Answer 3

Question

Chapter 14.7, Problem 64E

To determine

The point P(x, y) that minimize the sum of distances between the fixed points lying in a plane.

To determine

(a)

To Show:

That the condition $\nabla f (P) = 0$ is equivalent to $e + f + g = 0$ .

To determine

(b)

To Show:

That the function $f (x, y)$ is differentiable except at points A, B and C and the minimum of $f (x, y)$ occurs either at P satisfying the equation $e + f + g = 0$ , where e, f and g are the unit vectors in the direction of vectors $\vec{P A}, \vec{P B}$ and $\vec{P C}$ .

To determine

(c)

To Prove:

That the equation $e + f + g = 0$ , where e, f and g are the unit vectors in the direction of vectors holds if and only if P is the Fermat point, defined as the point P for which the angles between the segments $\bar{A P}, \bar{B P}$ and $\bar{C P}$ are all 120°.

To determine

(d)

To Show:

That the Fermat point does not exist if one of the angles in $Δ A B C$ is greater than 120° and the point which exist in this case is different from Fermat point.

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Answer 4

Question

Chapter 14.7, Problem 64E

To determine

The point P(x, y) that minimize the sum of distances between the fixed points lying in a plane.

To determine

(a)

To Show:

That the condition $\nabla f (P) = 0$ is equivalent to $e + f + g = 0$ .

To determine

(b)

To Show:

That the function $f (x, y)$ is differentiable except at points A, B and C and the minimum of $f (x, y)$ occurs either at P satisfying the equation $e + f + g = 0$ , where e, f and g are the unit vectors in the direction of vectors $\vec{P A}, \vec{P B}$ and $\vec{P C}$ .

To determine

(c)

To Prove:

That the equation $e + f + g = 0$ , where e, f and g are the unit vectors in the direction of vectors holds if and only if P is the Fermat point, defined as the point P for which the angles between the segments $\bar{A P}, \bar{B P}$ and $\bar{C P}$ are all 120°.

To determine

(d)

To Show:

That the Fermat point does not exist if one of the angles in $Δ A B C$ is greater than 120° and the point which exist in this case is different from Fermat point.

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Answer 5

Chapter 14.7, Problem 64E

To determine

The point P(x, y) that minimize the sum of distances between the fixed points lying in a plane.

To determine

(a)

To Show:

That the condition $\nabla f (P) = 0$ is equivalent to $e + f + g = 0$ .

To determine

(b)

To Show:

That the function $f (x, y)$ is differentiable except at points A, B and C and the minimum of $f (x, y)$ occurs either at P satisfying the equation $e + f + g = 0$ , where e, f and g are the unit vectors in the direction of vectors $\vec{P A}, \vec{P B}$ and $\vec{P C}$ .

To determine

(c)

To Prove:

That the equation $e + f + g = 0$ , where e, f and g are the unit vectors in the direction of vectors holds if and only if P is the Fermat point, defined as the point P for which the angles between the segments $\bar{A P}, \bar{B P}$ and $\bar{C P}$ are all 120°.

To determine

(d)

To Show:

That the Fermat point does not exist if one of the angles in $Δ A B C$ is greater than 120° and the point which exist in this case is different from Fermat point.

Answer 6

Chapter 14.7, Problem 64E

To determine

The point P(x, y) that minimize the sum of distances between the fixed points lying in a plane.

Answer 7

Chapter 14.7, Problem 64E

To determine

The point P(x, y) that minimize the sum of distances between the fixed points lying in a plane.

Answer 8

To determine

(a)

To Show:

That the condition $\nabla f (P) = 0$ is equivalent to $e + f + g = 0$ .

Answer 9

To determine

(a)

To Show:

That the condition $\nabla f (P) = 0$ is equivalent to $e + f + g = 0$ .

Answer 10

To determine

(b)

To Show:

That the function $f (x, y)$ is differentiable except at points A, B and C and the minimum of $f (x, y)$ occurs either at P satisfying the equation $e + f + g = 0$ , where e, f and g are the unit vectors in the direction of vectors $\vec{P A}, \vec{P B}$ and $\vec{P C}$ .

Answer 11

To determine

(b)

To Show:

That the function $f (x, y)$ is differentiable except at points A, B and C and the minimum of $f (x, y)$ occurs either at P satisfying the equation $e + f + g = 0$ , where e, f and g are the unit vectors in the direction of vectors $\vec{P A}, \vec{P B}$ and $\vec{P C}$ .

Answer 12

To determine

(c)

To Prove:

That the equation $e + f + g = 0$ , where e, f and g are the unit vectors in the direction of vectors holds if and only if P is the Fermat point, defined as the point P for which the angles between the segments $\bar{A P}, \bar{B P}$ and $\bar{C P}$ are all 120°.

Answer 13

To determine

(c)

To Prove:

That the equation $e + f + g = 0$ , where e, f and g are the unit vectors in the direction of vectors holds if and only if P is the Fermat point, defined as the point P for which the angles between the segments $\bar{A P}, \bar{B P}$ and $\bar{C P}$ are all 120°.

Answer 14

To determine

(d)

To Show:

That the Fermat point does not exist if one of the angles in $Δ A B C$ is greater than 120° and the point which exist in this case is different from Fermat point.

Answer 15

To determine

(d)

To Show:

That the Fermat point does not exist if one of the angles in $Δ A B C$ is greater than 120° and the point which exist in this case is different from Fermat point.

Answer 16

Expert Solution & Answer

Answer 17

Expert Solution & Answer

Answer 18

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Answer 19

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

The point P ( x , y ) that minimize the sum of distances between the fixed points lying in a plane.

The point P(x, y) that minimize the sum of distances between the fixed points lying in a plane.

(a) To Show: That the condition ∇f(P)=0 is equivalent to e+f+g=0.

(b) To Show: That the function f(x,y) is differentiable except at points A, B and C and the minimum of f(x,y) occurs either at P satisfying the equation e+f+g=0, where e, f and g are the unit vectors in the direction of vectors PA→,PB→ and PC→.

(c) To Prove: That the equation e+f+g=0, where e, f and g are the unit vectors in the direction of vectors holds if and only if P is the Fermat point, defined as the point P for which the angles between the segments AP¯,BP¯ and CP¯ are all 120°.

(d) To Show: That the Fermat point does not exist if one of the angles in ΔABC is greater than 120° and the point which exist in this case is different from Fermat point.

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Chapter 14 Solutions

(a)

To Show:

That the condition $\nabla f (P) = 0$ is equivalent to $e + f + g = 0$ .

(b)

To Show:

That the function $f (x, y)$ is differentiable except at points A, B and C and the minimum of $f (x, y)$ occurs either at P satisfying the equation $e + f + g = 0$ , where e, f and g are the unit vectors in the direction of vectors $\vec{P A}, \vec{P B}$ and $\vec{P C}$ .

(c)

To Prove:

That the equation $e + f + g = 0$ , where e, f and g are the unit vectors in the direction of vectors holds if and only if P is the Fermat point, defined as the point P for which the angles between the segments $\bar{A P}, \bar{B P}$ and $\bar{C P}$ are all 120°.

(d)

To Show:

That the Fermat point does not exist if one of the angles in $Δ A B C$ is greater than 120° and the point which exist in this case is different from Fermat point.