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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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Applying the Second-Partials Test In Exercises 1-18, find the relative extrema and saddle points of the function. See Examples

f ( x , y ) = 3 e ( x 2 + y 2 )

Chapter 7.5, Problem 18E, Applying the Second-Partials Test In Exercises 1-18, find the relative extrema and saddle points of

To determine

To calculate: The relative extreme and saddle point of the function f(x,y)=3e(x2+y2)

Explanation

Given Information:

The equation of the function is given as f(x,y)=3e(x2+y2)

The provided figure is,

Formula used:

To check the relative extrema and saddle points

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

Following steps to be followed:

Step 1: If d>0 and fxx(a,b)>0 then the function has a relative minimum at (a, b).

Step 2: If d>0 and fxx(a,b)<0 then the function has a relative minimum at (a, b).

Step 3: If d<0 then (a,b,f(a,b)) is a saddle point.

Step 4: The best gives us information when d=0

Calculation:

Consider the given equation,

f(x,y)=3e(x2+y2)

Now, differentiate with respect to x as,

fx(x,y)={3×(2x)e(x2+y2)}fx(x,y)=6xe(x2+y2)

Now, differentiate with respect to y as,

fy(x,y)={3×(2y)e(x2+y2)}fy(x,y)=6ye(x2+y2)

So, fx(x,y) and fy(x,y) are defined for all points in the xy-plane

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