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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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BuyFindarrow_forward

Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
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

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Chapter 14 Solutions

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