Find the absolute maximum and minimum of the function fxyx2y

Find the absolute maximum and minimum of the function f(x,y)=x^2+y^2 subject to the constraint x^4+y^4=81. As usual, ignore unneeded answer blanks, and list points in lexicographic order.

Absolute minimum value:

attained at ( 0 , 9 ), ( , ), ( , ), ( , ).

Absolute maximum value:

attained at ( , ), ( , ), ( , ), ( , ).

Solution

given

f(x,y)=x^2+y^2 subject to the constraint x^4+y^4=81.

now find fx

fx = 2x =0

x =0

fy =2y =0

y =0

so critical point is(0,0)

f(x,y)=x^2+y^2 subject to the constraint x^4+y^4=81.

y^4 = 81 - x^2

y^2 = sqrt( 81 - x^2)

f(x,y) = x^2 + sqrt(81 -x^2)

f\'(x,y) = 2x + 1/2sqrt(81 -x^2) =0

so -2x = 1/2sqrt(81 -x^2)

squaring on both sides

4x^2 = 1/4 (81 -x^2)

4x^2 4(81 -x^2) =1

16x^2 (81 -x^2) =1

1296x^2 -16x^4 -1 =0

16x^4 -1296x^2 +1 =0

after solving we get (+/- 8.99995 , +/-0.027777)

now we have to find y value keeping

these values

y^4 = 81 - (8.99995)^4

we get some values in complex form so leave it

minimum value:

attained at ( 3 , -3), (-3 ,3 ), (-3 , -3), (3 ,3 ) min value is 18

maximum at (0,9) (9,0) (-9 ,0) (0,-9) and max value is 81