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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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True or False? In Exercises 55 and 56, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If f ( x , y ) has a relative maximum at ( x 0 , y 0 , z 0 ) , then f x ( x 0 , y 0 ) = f y ( x 0 , y 0 ) = 0

To determine

Whether the truth of the statement “if f(x,y) has a relative maximum at (x0,y0) then fx(x0,y0)=fy(x0,y0)=0.

Explanation

Given Information:

The statement “if f(x,y) has a relative maximum at (x0,y0) then fx(x0,y0)=fy(x0,y0)=0.

The gradient of a multivariable function at a maximum or minimum point will be the zero vector, which corresponds to the graph having a flat tangent plane.

So, for a multivariable function f(x,y) the critical point exists for,

fx=fx=0fy=fy=0

But this doesn’t say anything about the nature of the critical point found with the above procedure.

One needs to perform the second derivative test to determine the nature of the critical point. For that one needs to define the following function at the critical point (x0,y0).

D=fxx(x0,yo)fyy(x0,yo)[fxy(x0,yo)]2

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