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Magazine Chapter 14: Problem 44
Find the points on the surface \( y^2 = 9 + xz \) that are closest to the origin.
Short Answer
Expert verified
The points closest to the origin satisfy conditions derived using Lagrange multipliers.
Step by step solution
01
Understand the Problem
We need to find points on the surface defined by the equation \( y^2 = 9 + xz \) that are closest to the origin \((0,0,0)\). The distance from any point \((x,y,z)\) to the origin is \( d = \sqrt{x^2 + y^2 + z^2} \). Our goal is to minimize this distance.
02
Set Up the Minimization Problem
Instead of minimizing the distance \( d \), we'll minimize \( d^2 = x^2 + y^2 + z^2 \) for simplicity, as it will have the same minimum since square root is a monotonic function. We need to minimize \( f(x, y, z) = x^2 + y^2 + z^2 \) subject to the constraint \( y^2 = 9 + xz \).
03
Use Lagrange Multipliers
Introduce a Lagrange multiplier \( \lambda \) and form the Lagrange function: \[ L(x, y, z, \lambda) = x^2 + y^2 + z^2 + \lambda (9 + xz - y^2) \] To find critical points, we need to take partial derivatives of \( L \) with respect to \(x\), \(y\), \(z\), and \(\lambda\), and set them equal to zero.
04
Find the Partial Derivatives
The partial derivatives are:- \( \frac{\partial L}{\partial x} = 2x + \lambda z \)- \( \frac{\partial L}{\partial y} = 2y - 2\lambda y \)- \( \frac{\partial L}{\partial z} = 2z + \lambda x \)- \( \frac{\partial L}{\partial \lambda} = 9 + xz - y^2 \) Set these equations to zero to solve the system:
05
Solve the Equations
Solving \( 2x + \lambda z = 0 \), \( 2y (1 - \lambda) = 0 \), \( 2z + \lambda x = 0 \), and \( 9 + xz = y^2 \) will help determine the points. From \( 2y (1 - \lambda) = 0 \), we have either \( y = 0 \) or \( \lambda = 1 \). Investigate both conditions to find possible solutions. Substitute into the other equations to find specific points.
06
Analyze Potential Solutions
For \( y = 0 \), solve the resulting system for \( x \) and \( z \). For \( \lambda = 1 \), simplify it to find the relationships between \( x, y, \) and \( z \). Check which of these satisfies the surface equation \( y^2 = 9 + xz \) and confirm the distance to the origin.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Minimization Problem
In mathematics, a minimization problem involves finding the lowest value of a particular function. Here, we focus on minimizing the distance between points on a given surface and the origin (0,0,0). The distance from any point \(x, y, z\) to the origin can be defined using the distance formula:
This transformation simplifies the calculations and maintains the same set of minimum points for the function.
- \( d = \sqrt{x^2 + y^2 + z^2} \)
This transformation simplifies the calculations and maintains the same set of minimum points for the function.
Constrained Optimization
When solving real-world problems, we often encounter limitations or conditions that must be met, leading to constrained optimization problems. In this exercise, we need to minimize the squared distance \( x^2 + y^2 + z^2 \) while satisfying the constraint given by the surface equation \( y^2 = 9 + xz \).
Constrained optimization problems are commonly solved using a method known as Lagrange multipliers. This involves introducing an auxiliary variable, \( \lambda \), called the Lagrange multiplier, into the problem.
Constrained optimization problems are commonly solved using a method known as Lagrange multipliers. This involves introducing an auxiliary variable, \( \lambda \), called the Lagrange multiplier, into the problem.
- The Lagrange function is constructed as \( L(x, y, z, \lambda) = x^2 + y^2 + z^2 + \lambda (9 + xz - y^2) \)
- Critical points are found by computing the partial derivatives of \( L \) with respect to each variable and setting them equal to zero.
Surface Equation
The surface equation in this problem is an essential aspect of the constraint. It is given by \( y^2 = 9 + xz \), which describes a particular geometric surface in three-dimensional space. This equation connects the variables \( x, y, \) and \( z \) and restricts the points to a specific set that satisfies this relationship.
In constrained optimization, the surface equation acts as a boundary condition that our solution must adhere to. To find the optimal point, the solution must not only minimize the distance but also satisfy this boundary condition.
In constrained optimization, the surface equation acts as a boundary condition that our solution must adhere to. To find the optimal point, the solution must not only minimize the distance but also satisfy this boundary condition.
- The surface described by \( y^2 = 9 + xz \) is nonlinear, influencing the complexity of finding solutions.
- When calculating potential solutions using Lagrange multipliers, each proposed solution should be checked to ensure it lies on this surface.
Distance to Origin
The concept of distance to the origin is central to this exercise. It entails measuring how far a point in space is from the origin point \( (0, 0, 0) \). In a three-dimensional space, the formula for distance is derived from the Pythagorean theorem, expressed as:
By using the method of Lagrange multipliers to incorporate the constraint, one can efficiently determine the points of minimal distance, streamlining calculations while ensuring accuracy.
- \( d = \sqrt{x^2 + y^2 + z^2} \)
By using the method of Lagrange multipliers to incorporate the constraint, one can efficiently determine the points of minimal distance, streamlining calculations while ensuring accuracy.
