Evaluate f at the solution>>, and then state (on the basis of all the informatio

Question

Evaluate f at the solution>>, and then state (on the basis of all the information you ht absolute maximum and absolute minimum values of f subject to the constraint g (as a cor Note that some solutions may be complex (Maple uses i to denote Squareroot-3); you do not f there. Also note that it is often necessary to interpret Maple's output correctly, for response squarerootof (_Z^5+_Z-3) means the single squareroot of Z^5 + Z - 3 = 0, and squarerootof (-Z*4 of the squareroots of Z^4 + Z - 3 = 0. There is just one squareroot of Z^5 + Z- 3 = 0 since J(Z) -increasing function of Z. Can you see how we know there is morr than one squareroot of Z^4 + Use Lagrange Multipliers to find the extrema of f subject to the given constraint: f(x, y) = 2x^2 + xy- y^2 + y, 2x + 3y=1 f(x, y) = x^2 -4Xy + 4y^2, x^3 + y^2 = 1

Explanation / Answer

b)f=x2 -4xy +4y2,g=x2+y2-1

f=g

<2x-4y,-4x+8y>=<2x,2y>

2x-4y =2x ,-4x+8y=2y

=1-(2y/x) ,=(-2x/y)+4

1-(2y/x) =(-2x/y)+4

(2x/y)-(2y/x)=3

(2x/y)-(2y/x)=3 => pair of intersecting lines y=-2x, y=x/2

x2+y2=1

on solving x2+y2=1, y=-2x =>(x,y)=(1/5 ,-2/5),(-1/5 ,2/5)

on solving x2+y2=1, y=x/2 =>(x,y)=(-2/5,-1/5 ),(2/5,1/5)

f(1/5 ,-2/5)=5

f(-1/5 ,2/5)=5

f(-2/5,-1/5 )=0

f(2/5,1/5)=0

absolute maximum=5 ,absolute minimum=0

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