Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

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

Short Answer

Expert verified
The critical point of the function \(f(x, y)=(x y)^{2}\) is (0,0). The Second-Partials Test fails at this point.

Step by step solution

01

Calculate the Partial Derivatives

The critical points are found by setting the first partial derivatives equal to zero. Start by calculating the partial derivative of the function with respect to x, denoted as \(f_x\), and with respect to y, denoted as \(f_y\). The given function is \(f(x, y)=(x y)^{2}\). Hence, \(f_x=2xy^2\) and \(f_y=2x^2y\).
02

Find the Critical Points

After computing the partial derivatives, we need to find the points that will make both \(f_x\) and \(f_y\) equal to zero. That implies solving the equations \(2xy^2=0\) and \(2x^2y=0\). Solving these gives the singular solution, that is (x,y)=(0,0).
03

Apply the Second-Partials Test

In the next step, apply the Second-Partials Test using the second order partial derivatives denoted as \(f_{xx}\), \(f_{yy}\), and \(f_{xy}\). These are calculated as: \(f_{xx}=2y^2\), \(f_{yy}=2x^2\), and \(f_{xy}=4xy\). The Second-Partials Test uses the formula \(D=f_{xx}f_{yy} - (f_{xy})^2\) where D is used to determine the type of critical point. Calculate D at (0,0) gives D = 0.
04

Determine the Extrema or Failures

Substitute the critical points into the formula from the second partials test. If D>0 and \(f_{xx}<0\) or \(f_{yy}<0\), then the critical point is a local maximum. If D >0 and \(f_{xx}>0\) or \(f_{yy}>0\), then the critical point is a local minimum. If D<0, then it's a saddle point. If D=0, the test is inconclusive. Here, D is 0 at the point (0,0), therefore the Second-Partials Test fails at this point.

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 Vaia!

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

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. $$ \int_{-2}^{2} \int_{0}^{4-y^{2}} d x d y $$

After a change in marketing, the weekly profit of the firm in Exercise 35 is given by \(P=200 x_{1}+580 x_{2}-x_{1}^{2}-5 x_{2}^{2}-2 x_{1} x_{2}-7500\) Estimate the average weekly profit if \(x_{1}\) varies between 55 and 65 units and \(x_{2}\) varies between 50 and 60 units.

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).

Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the given points. $$ (-5,1),(1,3),(2,3),(2,5) $$

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. $$ (0.5,2),(0.75,1.75),(1,3),(1.5,3.2),(2,3.7),(2.6,4) $$

See all solutions

Recommended explanations on Math Textbooks

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.

Sign-up for free