   Chapter 14, Problem 10RQ

Chapter
Section
Textbook Problem

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.If (2, 1) is a critical point of f andfxx(2, 1)fyy(2, 1) < [fxy(2, 1)]2then f has a saddle point at (2, 1).

To determine

Whether the statement “If (2,1) is a critical point of f and fxx(2,1)fyy(2,1)<[fxy(2,1)]2 , then f has a saddle point at (2,1) ” is true or false.

Explanation

Given:

The function f has critical point at (2,1) and there occurs an inequality fxx(2,1)fyy(2,1)<[fxy(2,1)]2 .

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

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