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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.

6. f(x, y) = xy − 2x − 2yx2y2

To determine

To find: The local maximum, local minimum and saddle point of the

function f(x,y)=xy2x2yx2y2 .

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)=xy2x2yx2y2 .

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

fx=x(xy2x2yx2y2)=x(xy)x(2x)x(2y)x(x2)x(y2)=y202x0=y22x

Thus, fx=y22x . (1)

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

fy=y(xy2x2yx2y2)=y(xy)y(2x)y(2y)y(x2)y(y2)=x0202y=x22y

Thus, fy=x22y . (2)

Solve the equations (1) and (2) and find the values of x and y.

From the equation (1),

y22x=0y2=2xy=2x+2

Substitute the value y=2x+2 in the equation (2) and obtain the value ofx.

x22(2x+2)=0x24x4=03x6=0x=2

Substitute the value of x=2 in the equation y=2x+2 and obtain y value.

y=2(2)+2=4+2=2

Thus, the critical point is, (x,y)=(2,2)

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Chapter 14 Solutions

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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-37RECh-14 P-38RECh-14 P-39RECh-14 P-40RECh-14 P-41RECh-14 P-42RECh-14 P-43RECh-14 P-44RECh-14 P-45RECh-14 P-46RECh-14 P-47RECh-14 P-48RECh-14 P-49RECh-14 P-50RECh-14 P-51RECh-14 P-52RECh-14 P-53RECh-14 P-54RECh-14 P-55RECh-14 P-56RECh-14 P-57RECh-14 P-58RECh-14 P-59RECh-14 P-60RECh-14 P-61RECh-14 P-62RECh-14 P-63RECh-14 P-64RECh-14 P-65RECh-14 P-1PCh-14 P-2PCh-14 P-3PCh-14 P-4PCh-14 P-5PCh-14 P-6PCh-14 P-7PCh-14 P-8P

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