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CalculusCalculus: An Applied Approach (MindTap Course List)10th Edition

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Chapter 7, Problem 15TYS
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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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In Exercises 14 and 15, find the relative extrema and saddle points of the function.

f ( x , y )   = 4 x − x 4 − y 4

To determine

To calculate: The relative extrema and saddle points of the function f(x,y)=4xy−x4−y4

Explanation

Given Information:

The provided function is f(x,y)=4xy−x4−y4

Formula used:

Let f(x,y) be the real valued function of two variable and their second partial derivative exist.

Define d(x,y) to be determinant,

d(x,y)=fxx(x,y)fyy(x,y)−[fxy(x,y)]2.

Here, (a,b) is the critical point such that, fx(a,b)=0 & fy(a,b)=0

The second derivative test as follows,

1. If d>0 and fxx(a,b)>0→Relative minima exist at critical point (a,b)

2. If d>0 and fxx(a,b)<0→Relative maxima exist at critical point (a,b)

3. If d<0 then (a,b,f(a,b))→Saddle point of f

Calculation:

Consider the function, f(x,y)=4xy−x4−y4

Now first order partial derivative of the function f(x,y) with respect to x is

fx(x,y)=4y−4x3=0

And, first order partial derivative of the function f(x,y) with respect to y is

fy(x,y)=4x−4y3=0

Hence, the critical points are (1,1)(−1,−1) & (0,0)

Now, differentiate the equation fx(x,y)=4y−4x3=0 with respect to x.

fxx(x,y)=−12x2, fxy(x,y)=4

Now, differentiate the equation fy(x,y)=4x−4y3=0 with respect to y

fyy(x,y)=−12y2

Now evaluate the determinant, d=fxx(a,b)⋅fyy(a,b)−

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Sect-7.1 P-1CPSect-7.1 P-2CPSect-7.1 P-3CPSect-7.1 P-4CPSect-7.1 P-5CPSect-7.1 P-6CPSect-7.1 P-7CPSect-7.1 P-1SWUSect-7.1 P-2SWUSect-7.1 P-3SWU
Sect-7.1 P-4SWUSect-7.1 P-5SWUSect-7.1 P-6SWUSect-7.1 P-7SWUSect-7.1 P-8SWUSect-7.1 P-1ESect-7.1 P-2ESect-7.1 P-3ESect-7.1 P-4ESect-7.1 P-5ESect-7.1 P-6ESect-7.1 P-7ESect-7.1 P-8ESect-7.1 P-9ESect-7.1 P-10ESect-7.1 P-11ESect-7.1 P-12ESect-7.1 P-13ESect-7.1 P-14ESect-7.1 P-15ESect-7.1 P-16ESect-7.1 P-17ESect-7.1 P-18ESect-7.1 P-19ESect-7.1 P-20ESect-7.1 P-21ESect-7.1 P-22ESect-7.1 P-23ESect-7.1 P-24ESect-7.1 P-25ESect-7.1 P-26ESect-7.1 P-27ESect-7.1 P-28ESect-7.1 P-29ESect-7.1 P-30ESect-7.1 P-31ESect-7.1 P-32ESect-7.1 P-33ESect-7.1 P-34ESect-7.1 P-35ESect-7.1 P-36ESect-7.1 P-37ESect-7.1 P-38ESect-7.1 P-39ESect-7.1 P-40ESect-7.1 P-41ESect-7.1 P-42ESect-7.1 P-43ESect-7.1 P-44ESect-7.2 P-1CPSect-7.2 P-2CPSect-7.2 P-3CPSect-7.2 P-1SWUSect-7.2 P-2SWUSect-7.2 P-3SWUSect-7.2 P-4SWUSect-7.2 P-5SWUSect-7.2 P-6SWUSect-7.2 P-1ESect-7.2 P-2ESect-7.2 P-3ESect-7.2 P-4ESect-7.2 P-5ESect-7.2 P-6ESect-7.2 P-7ESect-7.2 P-8ESect-7.2 P-9ESect-7.2 P-10ESect-7.2 P-11ESect-7.2 P-12ESect-7.2 P-13ESect-7.2 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Show that the equation of the tangent plane to the elliptic paraboloid z/c = x2/a2 + y2/b2 at the point (x0, y0...

Multivariable Calculus

Solve each equation in Exercises 107120 for x, rounding your answer to four significant digits where necessary....

Applied Calculus

Sketching a Graph In Exercises 91-94, sketch the graph of the function. Use a graphing utility to verify your g...

Calculus: Early Transcendental Functions

What value is represented by the uppercase letter N? a. the number of scores in a population b. the number of s...

Essentials of Statistics for The Behavioral Sciences (MindTap Course List)

SW A large counseling center needs to evaluate several experimental programs. Write a paragraph summarizing the...

Essentials Of Statistics

In Exercises 6372, evaluate the expression. 63. |6 + 2|

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

Find f(a). f(x)=12x

Calculus (MindTap Course List)

Describe what is measured by the estimated standard error in the bottom of the independent-measures t statistic...

Statistics for The Behavioral Sciences (MindTap Course List)

Statistical Literacy A data set has values ranging from a low of 10 to a high of 52. What's wrong with using th...

Understanding Basic Statistics

If a rock is thrown upward on the planet Mars with a velocity of 10 m/s, its height in meters t seconds later i...

Single Variable Calculus

In problems 7-10, replace each radical with a fractional exponent.Do not simplify. 8.

Mathematical Applications for the Management, Life, and Social Sciences

Simplify the expressions in Exercises 97106. (xy)1/3(yx)2/3

Finite Mathematics

Inductive vs. Deductive Reasoning Determine whether the argument is an example of inductive reasoning or deduct...

Mathematical Excursions (MindTap Course List)

Three couples and two single individuals have been invited to an investment seminar and have agreed to attend. ...

Probability and Statistics for Engineering and the Sciences

For Problems 55-94, simplify each numerical expression. Objectives 7 25(34)(12)(35)

Intermediate Algebra

If a ball is thrown upward from the lop of a building 128 ft high with an initial velocity of 16 ft/s, then the...

Single Variable Calculus: Early Transcendentals

Calculate the net asset value and number of shares purchased for the following mutual funds. Round shares to th...

Contemporary Mathematics for Business & Consumers

Draw the graph of the line through the given point with the given slope: (3,5),m=2

Elementary Technical Mathematics

In Exercise, Solve the systems of equations in.

Elements Of Modern Algebra

Complete each of the following tables. Round all answers to the nearest tenth.

Trigonometry (MindTap Course List)

Let N be any point on side BC of the right triangle ABC. Find the upper and lower limits for the length of AN.

Elementary Geometry For College Students, 7e

Using the Integral Test In Exercises 3-22, confirm that the Integral Test can be applied to the series. Then us...

Calculus of a Single Variable

Using a Theorem In Exercises 61-64, (a) use Theorem 9.5 lo show that the sequence with the given nth term conve...

Calculus: Early Transcendental Functions (MindTap Course List)

In Exercises 712, refer to the accompanying figure. Which point has an x coordinate that is equal to zero?

Finite Mathematics for the Managerial, Life, and Social Sciences

Suppose that r=12 cm and h=15 cm in the right circular cylinder. Find the exact and approximate a lateral area....

Elementary Geometry for College Students

True or False: converges.

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

An iterated integral for the volume of the solid under z = x2 + y2 and above the shaded region D is:

Study Guide for Stewart's Multivariable Calculus, 8th

Rectangular-to-Spherical ConversionIn Exercises 4350, find an equation in spherical coordinates for the surface...

Calculus

Evaluating an Iterated Integral Using Technology In Exercises 71-76, use a computer algebra system to evaluate ...

Multivariable Calculus

Finding the Direction of Maximum IncreaseIn Exercises 57 and 58, the temperature in degrees Celsius on the surf...

Calculus (MindTap Course List)

Find the area of each of the segments ACE for Exercises 41 through 45.

Mathematics For Machine Technology

Find the vertex, focus and directrix of each parabola. (x+2)2=24(y1)

College Algebra (MindTap Course List)

Define the goal or purpose of the correlational research strategy and distinguish between a correlational study...

Research Methods for the Behavioral Sciences (MindTap Course List)

Military radar and missile detection systems are designed to warn a country of an enemy attack. A reliability q...

STATISTICS F/BUSINESS+ECONOMICS-TEXT

The Economist collects data each year on the price of a Big Mac in various countries around the world. A sample...

Introduction To Statistics And Data Analysis

A clinical researcher has developed a new test for measuring impulsiveness and would like to determine the vali...

Research Methods for the Behavioral Sciences (MindTap Course List)

Consider the following data for two variables, x and y. x 22 24 26 30 35 40 y 12 21 33 35 40 36 a. Develop an e...

Statistics for Business & Economics, Revised (MindTap Course List)

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