   Chapter 14.7, Problem 11E

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.11. f(x, y) = x3 − 3x + 3xy2

To determine

To find: The local maximum, local minimum and saddle point of the function f(x,y)=x33x+3xy2 .

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”.

Calculation:

The given function is, f(x,y)=x33x+3xy2 .

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

fx=x(x33x+3xy2)=x(x3)3x(x)+3x(xy2)=3x23(1)+3(y2)=3x23+3y2

Thus, fx=3x23+3y2 . (1)

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

fy=y(x33x+3xy2)=y(x3)3y(x)+3xy(y2)=00+3x(2y)=6xy

Thus, fy=6xy . (2)

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

From the equation (2),

6xy=0xy=0x=0,y=0

Substitute the value y=0 in the equation (1).

3x23+3(0)2=03x23=0x2=33x=±1

Substitute the value x=0 in the equation (1).

3(0)23+3y2=03y23=0y2=33y=±1

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

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

2fx2=x(3x23+3y2)=x(3x2)x(3)+x(3y2)=3(2x)0+0=6x

Hence, 2fx2=6x .

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

2fy2=y(6xy)=6xy(y)=6x(1)=6x

Hence, 2fy2=6x .

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

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

Convert the expressions in Exercises 8596 radical form. 22/3

Finite Mathematics and Applied Calculus (MindTap Course List)

Find the mean for the following set of scores: 2, 7, 9, 4, 5, 3, 0, 6

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

For any events A and B with P(B) 0, show that P(A|B) + P(A|B) = 1.

Probability and Statistics for Engineering and the Sciences

In Exercises 17 to 22, factor completely. 2ax2+3ax-35a

Elementary Geometry For College Students, 7e

Find the derivative of the function. y=(x+1x)5

Single Variable Calculus: Early Transcendentals

True or False: Every series must do one of these: converge absolutely, converge conditionally, or diverge.

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

Graph each equation. 3x5y=3(x5)

College Algebra (MindTap Course List) 