(a) Draw contours of f(x, y) = 2x + y for z = −7, −5, −3, −1, 1, 3, 5, 7. (b) On the same axes, graph the constraint x2 +y2 = 5. (c) Use the graph to approximate the points at which f has a maximum or a minimum value subject to the constraint x2 + y2 = 5. (d) Use Lagrange multipliers to find the maximum and minimum values of f(x, y) = 2x + y subject to x2 + y2 = 5.

Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 17EQ
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(a) Draw contours of f(x, y) = 2x + y for z = −7, −5, −3, −1, 1, 3, 5, 7.

(b) On the same axes, graph the constraint x2 +y2 = 5.

(c) Use the graph to approximate the points at which f has a maximum or a minimum value subject to the constraint x2 + y2 = 5.

(d) Use Lagrange multipliers to find the maximum and minimum values of f(x, y) = 2x + y subject to x2 + y2 = 5.

 

Expert Solution
Step 1

aConsider a given function fx,y=2x+y and constraints: x2+y2-5=0Now,the lagrange's equation will begx,y=2x+y+λx2+y2-5gxx,y=2+2λxgyx,y=1+2λygλx,y=x2+y2-5Now,put gxx,y=0, gyx,y=0 and gλx,y=0Further solving, then2+2λx=0x=-1λ1+2λy=0y=-12λ

Step 2

aNote that λ0, thenx2+y2-5=01λ2+14λ2-5=054λ2-5=0λ2=14λ=±12Forλ=12x=-2,y=-1Forλ=-12x=2,y=1At x,y=-2,-1, thenf-2,-1=-5At x,y=2,1, thenf2,1=5

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