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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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Think About It In Exercises 19–24, determine whether there is a relative maximum, a relative minimum, a saddle point, or insufficient information to determine the nature of the function f(x, y) at the critical point (x0, y0).

f x x ( x 0 , y 0 ) = 9 , f y y ( x 0 , y 0 ) = 6 , f x y ( x 0 , y 0 ) = 10

To determine

Whether the function f(x,y) at the critical point (x0,y0) has a relative maximum, a relative minimum, a saddle point or insufficient information to determine the nature of the function.

Explanation

Given Information:

f(x,y) has a critical point (x0,y0) and the second partial derivatives are given as:

fxx(x0,y0)=9;fyy(x0,y0)=6;fxy(x0,y0)=10

For a multivariate function f(x,y) has a critical point (a,b) which is given by

fx(a,b)=dfdx|(a,b)=0fy(a,b)=dfdy|(a,b)=0

Now to determine the relative maxima or minima of the above critical point the second partial derivative test is performed.

Define,

D=fxx(a,b)fyy(a,b)[fxy(a,b)]2

1. If D>0 and fxx(a,b)>0 then (a,b) is a relative minimum at the above critical point.

2. If D<0 and fyy(a,b)<0 then (a,b) is a relative maximum at the above critical point

3

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