Use Lagrange multipliers to find the maximum and minimum values. f(x,y)=(x^2)y,x^2+y^2=4

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section9.4: Linear Programming
Problem 1E
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Use Lagrange multipliers to find the maximum and minimum values.

f(x,y)=(x^2)y,x^2+y^2=4

Expert Solution
Step 1

Consider the given function fx,y=x2y and the constraint gx,y=x2+y2-4.

Note that, fxx,y=λgxx,y, and fyx,y=λgyx,y.

That implies,

fxx,y=λgxx,y2xy=λ2x  ..... 1fyx,y=λgyx,yx2=λ2y    ..... 2

From (1),

2xy=λ2xλ=y

From (2),

x2=λ2yλ=x22y

 

Step 2

That implies,

y=x22y2y2=x2x=2y

Substitute x=2y in x2+y2=4.

2y2+y2=42y2+y2=43y2=4y2=43y=±23

Substitute y=23 in x=2y.

x=223=223

Substitute y=-23 in x=2y.

x=2-23=-223

 

 

 

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