minimize Q 6x23y2 where xy9SolutionQ 6x2 3y2 xy 9 x 9y s

minimize Q= 6x^2+3y^2 where x+y=9

Q = 6x^2 + 3y^2

x+y = 9

x = 9-y

substituting value of x in Q

Q = 6(9-y)^2 + 3y^2 = 6(81+y^2 - 18y ) + 3y^2 = 486 + 6y^2 - 108y + 3y^2

Q = 9y^2 - 108y + 486

Q\' = 18y - 108

set Q\' = 0 to find the minimum value

18y - 108 = 0

18y = 108

y = 6

x + y = 9

x + 6 = 9

x = 3

hence, Q = 6(3)^2 + 3(6)^2 = 162