/*! 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 15 Examine the function for relativ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 15

Examine the function for relative extrema and saddle points. $$ f(x, y)=\frac{1}{2} x y $$

Short Answer

Expert verified
The function \(f(x, y) = \frac{1}{2}xy\) has a saddle point at the origin \((0,0)\).

Step by step solution

01

Find the first partial derivatives

Calculate the first partial derivatives of the function \(f(x, y)\) with respect to \(x\) and \(y\).\n\n Derivative with respect to \(x\): \(f_x = \frac{1}{2} y\)\n\n Derivative with respect to \(y\): \(f_y = \frac{1}{2} x\)
02

Find the critical points

Critical points occur where the first partial derivatives are both zero or undefined. However, neither \(f_x\) nor \(f_y\) are ever undefined in this case. Thus set both \(f_x\) and \(f_y\) to zero and solve for \(x\) and \(y\). This gives the point \((0,0)\).
03

Find the second partial derivatives

We calculate the second partial derivatives of the function \(f(x, y)\). We also need to calculate the mixed second-order derivative.\n\n Second derivative with respect to \(x\): \(f_{xx} = 0\)\n\n Second derivative with respect to \(y\): \(f_{yy} = 0\)\n\n Second derivative with respect to \(x\) and then \(y\): \(f_{xy} = \frac{1}{2}\)
04

Applied the second derivative test

To find out whether our critical point \((0,0)\) is a local maximum, minimum or saddle point, we apply the second derivative test. We need to evaluate the determinant of the Hessian matrix, \(D = f_{xx} f_{yy} - (f_{xy})^2\), at the critical point. However, due to the fact that both \(f_{xx}\) and \(f_{yy}\) are zero, the determinant \(D\) will always be negative, indicating that we have a saddle point at \((0,0)\).

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 double integral to find the area of the region bounded by the graphs of the equations. $$ y=x^{3 / 2}, y=x $$

The revenues \(y\) (in millions of dollars) for Earthlink from 2000 through 2006 are shown in the table. $$ \begin{aligned} &\begin{array}{|l|l|l|l|l|} \hline \text { Year } & 2000 & 2001 & 2002 & 2003 \\ \hline \text { Revenue, } y & 986.6 & 1244.9 & 1357.4 & 1401.9 \\ \hline \end{array}\\\ &\begin{array}{|l|l|l|l|} \hline \text { Year } & 2004 & 2005 & 2006 \\ \hline \text { Revenue, } y & 1382.2 & 1290.1 & 1301.3 \\ \hline \end{array} \end{aligned} $$ (a) Use a graphing utility or a spreadsheet to create a scatter plot of the data. Let \(t=0\) represent the year 2000 . (b) Use the regression capabilities of a graphing utility or a spreadsheet to find an appropriate model for the data. (c) Explain why you chose the type of model that you created in part (b).

Set up the integral for both orders of integration and use the more convenient order to evaluate the integral over the region \(R\). $$ \begin{aligned} &\int_{R} \int x y d A\\\ &R \text { : rectangle with vertices at }(0,0),(0,5),(3,5),(3,0) \end{aligned} $$

Use a double integral to find the volume of the solid bounded by the graphs of the equations. $$ z=x^{2}, z=0, x=0, x=2, y=0, y=4 $$

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

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Pure Maths

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.