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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Find the extreme values of f on the region described by the inequality.

23. f(x, y) = e-xy, x2 + 4y2 ≤ 1

To determine

To find: The extreme values of the function f(x,y)=exy on the disk x2+4y21 .

Explanation

Given:

The function is f(x,y)=exy on the disk x2+4y21 .

Definition used:

“The Lagrange multipliers defined as f(x,y,z)=λg(x,y,z) . This equation can be expressed as fx=λgx , fy=λgy , fz=λgz and g(x,y,z)=k ”.

Calculation:

Find the partial derivatives of the function f(x,y)=exy is computed as follows,

f(x,y)=exyfx(x,y)=yexyfy(x,y)=xexy

Set the partial derivatives fx(x,y)=yexy,fy(x,y)=xexy to zero and find the values of xandy .

fx(x,y)=yexy=0x=0fy(x,y)=xexy=0y=0

Thus, the critical point of the function f(x,y)=exy is (0,0) .

This critical point (1,0) is inside the given disk x2+4y21 as it satisfies the inequality.

Let the boubndary function of the inequality x2+4y21 be g(x,y)=x2+4y2=1 .

The Lagrange multipliers f(x,y)=λg(x,y) is computed as follows,

f(x,y)=λg(x,y)fx,fy=λgx,gyfx(exy),fy(exy)=λgx(x2+4y2),gy(x2+4y2)yexy,xexy=λ2x,8y

Thus, the value of f(x,y)=λg(x,y) is yexy,xexy=λ2x,8y

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