   Chapter 14.7, Problem 12E

Chapter
Section
Textbook Problem

Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function.12. f(x, y) = x3 + y3 − 3x2 − 3y2 − 9x

To determine

To find: The local maximum, local minimum and saddle point of the function f(x,y)=x3+y33x23y29x .

Explanation

Result used:

Second Derivative Test:

“Suppose the second partial derivatives of f are continuous on a disk with center (a,b) , and suppose that fx(a,b)=0 and fy(a,b)=0 (that is (a,b) is a critical point of f).

Let D=D(a,b)=fxx(a,b)fyy(a,b)[fxy(a,b)]2

(a) If D>0 and fxx(a,b)>0 , then f(a,b) is a local minimum.

(b) If D>0 and fxx(a,b)<0 , then f(a,b) is a local maximum.

(c) If D<0 , then f(a,b) is not a local maximum or minimum and it is called a saddle point”.

Given:

The function is, f(x,y)=x3+y33x23y29x .

Calculation:

Take the partial derivative in the given function with respect to x and obtain fx .

fx=x(x3+y33x23y29x)=x(x3)+x(y3)3x(x2)3x(y2)9x(x)=3x2+03(2x)09(1)=3x26x9

Thus, fx=3x26x9 . (1)

Take the partial derivative with respect to y and obtain fy .

fy=y(x3+y33x23y29x)=y(x3)+y(y3)y(3x2)y(3y2)9y(x)=0+3y203(2y)0=3y26y

Thus, fy=3y26y (2)

Set the above partial derivatives to 0 and find the values of x and y.

From the equation (2),

3y26y=03y(y2)=03y=0,y2=0y=0,y=2

From the equation (1),

3x26x9=03(x22x3)=03(x+1)(x3)=0x=1,3

Thus, the critical points are, (1,0) , (1,2) , (3,0) and (3,2) .

Take the partial derivative of the equation (1) with respect to x and obtain fxx .

2fx2=x(3x26x9)=3x(x2)6x(x)x(9)=3(2x)6(1)0=6x6

Hence, 2fx2=6x6 .

Take the partial derivative of the equation (2) with respect to y and obtain fyy .

2fy2=y(3y26y)=3y(y2)6y(y)=3(2y)6(1)=6y6

Hence, 2fy2=6y6 .

Take the partial derivative of the equation (1) with respect to y and obtain fxy .

2fxy=y(3x26x9)=y(3x2)y(6x)y(9)=0

Hence, 2fxy=0

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

Find the indicated power using De Moivres Theorem. 34. (13i)5

Single Variable Calculus: Early Transcendentals, Volume I

In Problems 39 – 44, solve each inequality and graph the solution. 44.

Mathematical Applications for the Management, Life, and Social Sciences

limx6x33+x= a) b) c) 0 d) 2

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