/*! 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 2 Find any critical points and rel... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 2

Find any critical points and relative extrema of the function. $$ f(x, y)=x^{2}+y^{2}+2 x-6 y+6 $$

Short Answer

Expert verified
The function has one critical point at (-1,3), and this point is a relative minimum.

Step by step solution

01

Find the Gradient of the Function

The gradient of the function, which is the vector of all first partial derivatives, is found by taking the derivative of \(f(x, y)\) with respect to each variable. For \(f(x, y)=x^{2}+y^{2}+2 x-6 y+6\), the gradient of f is \(\nabla f = (2x + 2, 2y - 6)\)
02

Find the Critical Points

Now set the gradient obtained in the previous step to zero and solve for x and y. This gives us the equations \(2x + 2 = 0\) and \(2y - 6 = 0\). Solving these gives \(x = -1\) and \(y = 3\), so there's one critical point at (-1, 3).
03

Use the Second Derivative Test

The second derivative test determines whether the critical point is a maximum, a minimum, or a saddle point. It involves finding the Hessian matrix (the matrix of all second partial derivatives) and calculating its determinant. The second-order partial derivatives of \(f(x, y)\) are all constant: \(f_{xx} = 2\), \(f_{xy} = f_{yx} = 0\), \(f_{yy} = 2\). The determinant of the Hessian matrix, often called 'D', is \(D = f_{xx} f_{yy} - (f_{xy})^2 = (2)(2) - (0)^2 = 4\). Because \(D > 0\) and \(f_{xx} > 0\), the point (-1,3) is a relative minimum.

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

Evaluate the partial integral. $$ \int_{x}^{x^{2}} \frac{y}{x} d y $$

Use the regression capabilities of \(a\) graphing utility or a spreadsheet to find any model that best fits the data points. $$ \begin{aligned} &(1,13), \quad(2,16.5),(4,24),(5,28),(8,39),(11,50.25) \\ &(17,72),(20,85) \end{aligned} $$

The global numbers of personal computers \(x\) (in millions) and Internet users \(y\) (in millions) from 1999 through 2005 are shown in the table. $$ \begin{aligned} &\begin{array}{|l|c|c|c|c|} \hline \text { Year } & 1999 & 2000 & 2001 & 2002 \\ \hline \text { Personal computers, } x & 394.1 & 465.4 & 526.7 & 575.5 \\ \hline \text { Internet users, } y & 275.5 & 390.3 & 489.9 & 618.4 \\ \hline \end{array}\\\ &\begin{array}{|l|c|c|c|} \hline \text { Year } & 2003 & 2004 & 2005 \\ \hline \text { Personal computers, } x & 636.6 & 776.6 & 808.7 \\ \hline \text { Internet users, } y & 718.8 & 851.8 & 982.5 \\ \hline \end{array} \end{aligned} $$ (a) Use a graphing utility or a spreadsheet to create a scatter plot of the data. (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).

Use a double integral to find the volume of the solid bounded by the graphs of the equations. $$ z=x y, z=0, y=0, y=4, x=0, x=1 $$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Decision 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.