/*! 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 26 Find the critical points and tes... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 26

Find the critical points and test for relative extrema. List the critical points for which the Second-Partials Test fails. $$ f(x, y)=\sqrt{x^{2}+y^{2}} $$

Short Answer

Expert verified
The function \(f(x, y) = \sqrt{x^{2}+y^{2}}\) has a critical point at (0,0), but the Second-Partials Test fails at this point because the second partial derivatives are undefined.

Step by step solution

01

Compute the first derivatives

First, compute the first partial derivatives of the function \(f(x, y)\) with respect to \(x\) and \(y\). The partial derivative with respect to \(x\) is given by \(f_{x}(x,y) = x/ \sqrt{x^{2}+y^{2}}\). The partial derivative with respect to \(y\) is \(f_{y}(x,y) = y/ \sqrt{x^{2}+y^{2}}\).
02

Find the critical points

The critical points are where the first partial derivatives are equal to zero or undefined. For \(f_x(x, y)\) and \(f_y(x, y)\), both are undefined at (0,0), which is the only critical point of this function.
03

Compute the second derivatives

Next, compute the second partial derivatives of the function. The second partial derivative with respect to \(x\) is given by \(f_{xx}(x,y) = y^{2}/(x^{2}+y^{2})^{3/2}\), and the one with respect to \(y\) gives \(f_{yy}(x,y)= x^{2}/(x^{2}+y^{2})^{3/2}\). The mixed second partial derivatives are \(f_{xy}(x,y)= f_{yx}(x,y)= -xy/(x^{2}+y^{2})^{3/2}\).
04

Apply the Second-Partials Test

The Second-Partials Test requires computing the determinant of the Hessian matrix, \(D(x, y) = f_{xx}f_{yy} - (f_{xy})^2\). At the critical point (0,0), all second partial derivatives are undefined, and thus, the Second-Partials test cannot be applied.

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

Use a symbolic integration utility to evaluate the double integral. $$ \int_{0}^{1} \int_{0}^{2} e^{-x^{2}-y^{2}} d x d y $$

Evaluate the partial integral. $$ \int_{1}^{e} \frac{y \ln x}{x} d x $$

Use a symbolic integration utility to evaluate the double integral. $$ \int_{0}^{3} \int_{0}^{x^{2}} \sqrt{x} \sqrt{1+x} d y d x $$

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. $$ (1,7.5),(2,7),(3,7),(4,6),(5,5),(6,4.9) $$

Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the given points. $$ (-10,10),(-5,8),(3,6),(7,4),(5,0) $$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

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.