/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 35 Find three positive numbers \(x,... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 35

Find three positive numbers \(x, y\), and \(z\) that satisfy the given conditions. The sum is 30 and the sum of the squares is a minimum.

Short Answer

Expert verified
The three positive numbers are \(x = 10\), \(y = 10\), and \(z = 10\)

Step by step solution

01

Write Down the Equations

We have two equations to work with, \(x + y + z = 30\) from the sum condition and \(f(x,y,z) = x^2 + y^2 + z^2\) for the minimum sum of the squares. We will try to minimize \(f(x,y,z)\).
02

Conversion to a Function of One Variable

Rewrite the equation in terms of one variable, this simplifies the problem. From the sum condition, we have \(z = 30 - x - y\). Substituting \(z\) into \(f(x,y,z)\), we get a new function, \(g(x,y) = x^2 + y^2 + (30 - x - y)^2\).
03

Find the partial derivatives and solve them

Find the partial derivatives of \(g(x,y)\) with respect to \(x\) and \(y\). Then, set these resulting equations equal to zero and solve them to get the critical points. If done correctly, one should find \(x = y = 10\).
04

Substitute Back the Results

Substitute \(x = 10\) and \(y = 10\) into the sum condition \(x + y + z = 30\) to get \(z = 10\).
05

Confirm that it minimize the function

To confirm that these values indeed minimize the function \(g(x,y)\) and don't maximize it or represent saddle points, we can use the Second Partial Derivative Test or reconsider the symmetry of \(g(x,y)\) in combination with its boundedness. As this function is symmetric and bounded from below by zero, and our critical point is a single point within the area of definition, it indeed has to be a global minimum. However, a rigorous proof would use the Second Partial Derivative Test in a formal manner.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Plot the points and determine whether the data have positive, negative, or no linear correlation (see figures below). Then use a graphing utility to find the value of \(r\) and confirm your result. The number \(r\) is called the correlation coefficient. It is a measure of how well the model fits the data. Correlation coefficients vary between \(-1\) and 1, and the closer \(|r|\) is to 1, the better the model. $$ (1,4),(2,6),(3,8),(4,11),(5,13),(6,15) $$

The Cobb-Douglas production function for an automobile manufacturer is \(f(x, y)=100 x^{0.6} y^{0.4}\) where \(x\) is the number of units of labor and \(y\) is the number of units of capital. Estimate the average production level if the number of units of labor \(x\) varies between 200 and 250 and the number of units of capital \(y\) varies between 300 and 325 .

Evaluate the double integral. Note that it is necessary to change the order of integration. $$ \int_{0}^{3} \int_{y}^{3} e^{x^{2}} d x d y $$

Exercises 55 and 56, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ \int_{2}^{5} \int_{1}^{6} x d y d x=\int_{1}^{6} \int_{2}^{5} x d x d y $$

Sketch the region \(R\) whose area is given by the double integral. Then change the order of integration and show that both orders yield the same area. $$ \int_{0}^{2} \int_{x / 2}^{1} d y d x $$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.