   Chapter 14.7, Problem 21E

Chapter
Section
Textbook Problem

Show that f(x, y) = x2 + 4y2 − 4xy + 2 has an infinite number of critical points and that D = 0 at each one. Then show that f has a local (and absolute) minimum at each critical point.

To determine

To find: The local maximum, local minimum and saddle point of the function f(x,y)=x2+4y24xy+2 .

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)=x2+4y24xy+2 .

Calculation:

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

fx=x(x2+4y24xy+2)=x(x2)+4x(y2)4x(xy)+x(2)=2x+04(y)+0=2x4y

Thus, fx=2x4y . (1)

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

fy=y(x2+4y24xy+2)=y(x2)+4y(y2)4y(xy)+y(2)=0+4(2y)4(x)+0=8y4x

Thus, fy=8y4x

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