Find the absolute maximum and minimum of the function f(x,y) = x^2-y^2 subject t

Question

Find the absolute maximum and minimum of the function f(x,y) = x^2-y^2 subject to the constraint x^2 +y^2=64. As usual, ignore unneeded answer blanks, and list points in lexicographic order. Absolute minimum value: Attained at( ,_)and(?-??-?)? Absolute maximum value: attained at ( ?) and (_____,............). Answer (s) submitted: Find the absolute maximum and minimum of the function f(x,y) = x^2 +y^2 subject to the constraint x^4+y^4 = 16. As usual ignore unneeded answer blanks, and list points in lexicographic order. Absolute minimum value: ______ attained at Absolute maximum value: attained at Answer(s) submitted:

Explanation / Answer

1) f(x,y) = x2 - y2 ; x2 +y2 = 64

==> g(x,y) = x2 +y2 -64

using lagrange multipliers

f(x,y) = g(x,y)

==> <fx ,fy> = <gx ,gy>

==> <2x , -2y> = <2x ,2y>

==> 2x = (2x) ; -2y = 2y

==> 2x(1 - ) = 0 ; 2y( +1) = 0

==> x = 0 , y = 0

x = 0 ==> (0)2 +y2 = 64

==> y = +8 , -8

y = 0 ==> x2 + 02 = 64

==> x = +8 , -8

Hence critical points are (0 ,8),(0,-8),(8,0),(-8,0)

f(0,8) = 02 - 82 = -64

f(0, -8) = 02 - (-8)2 = -64

f(8,0) = 82 - 02 = 64

f(-8,0) = (-8)2 - 02 = 64

Hence minimum at (0,8) & (0,-8) and value of function = -64

maximum at (8,0) & (-8,0) and value of function = 64

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