BuyFindarrow_forward

Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

Solutions

Chapter
Section
BuyFindarrow_forward

Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

In Problems 1-16, find each function’s relative maxima, relative minima, and saddle points, if they exist.

z = x 2 y 2 + 4 x 6 y + 11

To determine

To calculate: The relative maxima, relative minima, and saddle points of z=x2y2+4x6y+11, if they exist.

Explanation

Given Information:

The provided function is z=x2y2+4x6y+11.

Formula used:

Differentiate z=f(x,y) with respect to x by holding y constant as:

x[f(x,y)]=fx(x,y)

Differentiate f(x,y) with respect to y by holding x constant as:

y[f(x,y)]=fy(x,y)

The every differentiable function f(x,y),

2yxf(x,y)=2xyf(x,y)

The following procedure are used to calculate relative maximum and saddle point of given function f(x,y).

Step-1: Calculate first derivative of function f(x,y) with respect to x and y.

Step-2 Calculate second derivative of function f(x,y) with respect to x and y.

Step-3: Equate first derivative of function f(x,y) with respect to x to zero.

Step-4: Equate first derivative of function f(x,y) with respect to y to zero.

Step-5: Calculate critical point of function f(x,y) that is (a,b).

Step-6: Now test function f(x,y) at critical point (a,b) for extrema and saddle point that is describe in following table.

Critical point 22xf(x,y) 2x2f(x,y)2y2f(x,y)[2yxf(x,y)]2 Conclusion
(a,b) 22xf(a,b)>0 2x2f(a,b)2y2f(a,b)[2yxf(a,b)]2>0 The function f(x,y) has relative minimum at point (a,b)
(a,b) 22xf(a,b)<0 2x2f(a,b)2y2f(a,b)[2yxf(a,b)]2>0 The function f(x,y) has relative maximum at point (a,b)
(a,b) 2x2f(a,b)2y2f(a,b)[2yxf(a,b)]2<0 The function f(x,y) has saddle point (a,b,f(a,b))
(a,b) 2x2f(a,b)2y2f(a,b)[2yxf(a,b)]2=0 The test gives no information.

The simple power rule of derivative:

ddx(xn)=nxn1

The constant rule of derivative:

ddx(c)=0

The derivative of constant coefficient:

ddx(cf(x))=cf(x)

Calculation:

Consider the function, z=x2y2+4x6y+11.

Differentiate z=f(x,y) with respect to x by holding y constant,

zx=x(x2y2+4x6y+11)=2x+4

Equate the first derivative of function z=x2y2+4x6y+11 with respect to x to zero

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Chapter 14 Solutions

Show all chapter solutions add
Sect-14.1 P-10ESect-14.1 P-11ESect-14.1 P-12ESect-14.1 P-13ESect-14.1 P-14ESect-14.1 P-15ESect-14.1 P-16ESect-14.1 P-17ESect-14.1 P-18ESect-14.1 P-19ESect-14.1 P-20ESect-14.1 P-21ESect-14.1 P-22ESect-14.1 P-23ESect-14.1 P-24ESect-14.1 P-25ESect-14.1 P-27ESect-14.1 P-28ESect-14.1 P-29ESect-14.1 P-30ESect-14.1 P-31ESect-14.1 P-32ESect-14.1 P-33ESect-14.1 P-34ESect-14.1 P-35ESect-14.1 P-36ESect-14.1 P-37ESect-14.1 P-38ESect-14.2 P-1CPSect-14.2 P-2CPSect-14.2 P-3CPSect-14.2 P-4CPSect-14.2 P-5CPSect-14.2 P-1ESect-14.2 P-2ESect-14.2 P-3ESect-14.2 P-4ESect-14.2 P-5ESect-14.2 P-6ESect-14.2 P-7ESect-14.2 P-8ESect-14.2 P-9ESect-14.2 P-10ESect-14.2 P-11ESect-14.2 P-12ESect-14.2 P-13ESect-14.2 P-14ESect-14.2 P-15ESect-14.2 P-16ESect-14.2 P-17ESect-14.2 P-18ESect-14.2 P-19ESect-14.2 P-20ESect-14.2 P-21ESect-14.2 P-22ESect-14.2 P-23ESect-14.2 P-24ESect-14.2 P-25ESect-14.2 P-26ESect-14.2 P-27ESect-14.2 P-28ESect-14.2 P-29ESect-14.2 P-30ESect-14.2 P-31ESect-14.2 P-32ESect-14.2 P-33ESect-14.2 P-34ESect-14.2 P-35ESect-14.2 P-36ESect-14.2 P-37ESect-14.2 P-38ESect-14.2 P-39ESect-14.2 P-40ESect-14.2 P-41ESect-14.2 P-42ESect-14.2 P-43ESect-14.2 P-44ESect-14.2 P-45ESect-14.2 P-46ESect-14.2 P-47ESect-14.2 P-48ESect-14.2 P-49ESect-14.2 P-50ESect-14.2 P-51ESect-14.2 P-52ESect-14.2 P-53ESect-14.2 P-54ESect-14.2 P-55ESect-14.2 P-56ESect-14.3 P-1CPSect-14.3 P-2CPSect-14.3 P-3CPSect-14.3 P-1ESect-14.3 P-2ESect-14.3 P-3ESect-14.3 P-4ESect-14.3 P-5ESect-14.3 P-6ESect-14.3 P-7ESect-14.3 P-8ESect-14.3 P-9ESect-14.3 P-10ESect-14.3 P-11ESect-14.3 P-12ESect-14.3 P-13ESect-14.3 P-14ESect-14.3 P-15ESect-14.3 P-16ESect-14.3 P-17ESect-14.3 P-18ESect-14.3 P-19ESect-14.3 P-20ESect-14.3 P-21ESect-14.3 P-22ESect-14.3 P-23ESect-14.3 P-24ESect-14.3 P-25ESect-14.3 P-26ESect-14.3 P-27ESect-14.3 P-28ESect-14.3 P-29ESect-14.3 P-30ESect-14.4 P-1CPSect-14.4 P-2CPSect-14.4 P-3CPSect-14.4 P-4CPSect-14.4 P-1ESect-14.4 P-2ESect-14.4 P-3ESect-14.4 P-4ESect-14.4 P-5ESect-14.4 P-6ESect-14.4 P-7ESect-14.4 P-8ESect-14.4 P-9ESect-14.4 P-10ESect-14.4 P-11ESect-14.4 P-12ESect-14.4 P-13ESect-14.4 P-14ESect-14.4 P-15ESect-14.4 P-16ESect-14.4 P-17ESect-14.4 P-18ESect-14.4 P-19ESect-14.4 P-20ESect-14.4 P-21ESect-14.4 P-22ESect-14.4 P-23ESect-14.4 P-24ESect-14.4 P-25ESect-14.4 P-26ESect-14.4 P-27ESect-14.4 P-28ESect-14.4 P-29ESect-14.4 P-30ESect-14.4 P-31ESect-14.4 P-32ESect-14.4 P-34ESect-14.4 P-35ESect-14.4 P-36ESect-14.5 P-1CPSect-14.5 P-2CPSect-14.5 P-3CPSect-14.5 P-4CPSect-14.5 P-1ESect-14.5 P-2ESect-14.5 P-3ESect-14.5 P-4ESect-14.5 P-5ESect-14.5 P-6ESect-14.5 P-7ESect-14.5 P-8ESect-14.5 P-9ESect-14.5 P-10ESect-14.5 P-11ESect-14.5 P-12ESect-14.5 P-13ESect-14.5 P-14ESect-14.5 P-15ESect-14.5 P-16ESect-14.5 P-17ESect-14.5 P-18ESect-14.5 P-19ESect-14.5 P-20ESect-14.5 P-21ESect-14.5 P-22ESect-14.5 P-23ESect-14.5 P-24ESect-14.5 P-25ESect-14.5 P-26ECh-14 P-1RECh-14 P-2RECh-14 P-3RECh-14 P-4RECh-14 P-5RECh-14 P-6RECh-14 P-7RECh-14 P-8RECh-14 P-9RECh-14 P-10RECh-14 P-11RECh-14 P-12RECh-14 P-13RECh-14 P-14RECh-14 P-15RECh-14 P-16RECh-14 P-17RECh-14 P-18RECh-14 P-19RECh-14 P-20RECh-14 P-21RECh-14 P-22RECh-14 P-23RECh-14 P-24RECh-14 P-25RECh-14 P-26RECh-14 P-27RECh-14 P-28RECh-14 P-29RECh-14 P-30RECh-14 P-31RECh-14 P-32RECh-14 P-33RECh-14 P-34RECh-14 P-35RECh-14 P-36RECh-14 P-1TCh-14 P-2TCh-14 P-3TCh-14 P-4TCh-14 P-5TCh-14 P-6TCh-14 P-7TCh-14 P-8TCh-14 P-9TCh-14 P-10T

Additional Math Solutions

Find more solutions based on key concepts

Show solutions add

Postal Regulations In 2012, the postage for parcels sent by first-class mail was raised to 1.95 for any parcel ...

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

Convert the expressions in Exercises 31-36 to positive exponent form. 45y3/4

Finite Mathematics and Applied Calculus (MindTap Course List)

Evaluate the integral. 23(x23)dx

Calculus: Early Transcendentals

Solve each equation and check: 3n+(2n+4)=6

Elementary Technical Mathematics

What does the Chebychevs inequality measure?

Finite Mathematics for the Managerial, Life, and Social Sciences

Express the number as a ratio of integers. 54. 10.135=10.135353535

Single Variable Calculus: Early Transcendentals

Evaluate . e + 1

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