Chapter 12: Q. 58 (page 986)

Prove that if you minimize the square of the distance from the origin to a point (x, y) subject to the constraint $g (x, y) = 0$ , you have minimized the distance from the origin to (x, y) subject to the same constraint.

Short Answer

Expert verified

$d = \sqrt{x^{2} + y^{2}}, \nabla d (x, y) = (\frac{x}{\sqrt{x^{2} + y^{2}}}) i + (\frac{y}{\sqrt{x^{2} + y^{2}}}) j \nabla g (x, y) = 0 . S o l v i n g w e g e t m i n i m u m a t (0, 0) . D = x^{2} + y^{2}, \nabla D (x, y) = (2 x) i + (2 y) j \nabla g (x, y) = 0 . S o l v i n g w e g e t m i n i m u m a t (0, 0) . T h e r e f o r e, i n b o t h c a s e s w e g e t t h e s a m e s o l u t i o n .$

Step by step solution

Step 1. Given Information.

Given the constraint: $g (x, y) = 0 .$

And the point (x, y) is given on the plane.

Step 2. Minimize the distance from origin to (x, y).

Let d be the distance from origin to the point.

$d = \sqrt{x^{2} + y^{2}} .$

$\nabla d (x, y) = (\frac{\partial d}{\partial x}) i + (\frac{\partial d}{\partial y}) j \nabla d (x, y) = (\frac{x}{\sqrt{x^{2} + y^{2}}}) i + (\frac{y}{\sqrt{x^{2} + y^{2}}}) j A n d g i v e n, \nabla g (x, y) = 0 .$

So, by Lagrange's method:

localid="1649910485059" $\nabla d = λ \nabla g, (\frac{x}{\sqrt{x^{2} + y^{2}}}) i + (\frac{y}{\sqrt{x^{2} + y^{2}}}) j = λ (0 i + 0 j), T h i s i m p l i e s, \frac{x}{\sqrt{x^{2} + y^{2}}} = 0 a n d \frac{y}{\sqrt{x^{2} + y^{2}}} = 0 W h i c h i s e q u a l t o x = 0 a n d y = 0 . T h e r e f o r e, t h e m i n i m u m o c c u r s a t (0, 0) .$

Step 2. Minimize the square of the distance from origin to (x, y).

Let D be the distance from origin to the point.

$D = x^{2} + y^{2} . \nabla D (x, y) = (\frac{\partial D}{\partial x}) i + (\frac{\partial D}{\partial y}) j \nabla D (x, y) = (2 x) i + (2 y) j A n d g i v e n, \nabla g (x, y) = 0 .$

So, by Lagrange's method:

$\nabla D = λ \nabla g, (2 x) i + (2 y) j = λ (0 i + 0 j), T h i s i m p l i e s, 2 x = 0 a n d 2 y = 0 W h i c h i s e q u a l t o x = 0 a n d y = 0 . T h e r e f o r e, t h e m i n i m u m o c c u r s a t (0, 0) .$

Therefore, in both the cases the solution is the same.

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Prove that if you minimize the square of the distance from the origin to a point (x, y) subject to the constraint $g (x, y) = 0$ , you have minimized the distance from the origin to (x, y) subject to the same constraint.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Minimize the distance from origin to (x, y).

Step 2. Minimize the square of the distance from origin to (x, y).

One App. One Place for Learning.

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Prove that if you minimize the square of the distance from the origin to a point (x, y) subject to the constraint g(x,y)=0, you have minimized the distance from the origin to (x, y) subject to the same constraint.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Minimize the distance from origin to (x, y).

Step 2. Minimize the square of the distance from origin to (x, y).

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Probability and Statistics

Theoretical and Mathematical Physics

Applied Mathematics

Discrete Mathematics

Logic and Functions

Study anywhere. Anytime. Across all devices.

Prove that if you minimize the square of the distance from the origin to a point (x, y) subject to the constraint $g (x, y) = 0$ , you have minimized the distance from the origin to (x, y) subject to the same constraint.