Question

The function f(x,y) 4x yhas an absolute maximum value and absolute minimum value subject to the constraint x3y = 40. Use Lagrange multipliers to find these values.

Answer 1

1 e max or min froux) 2 2 sit. x² + 3y + 4 = 40 to this find the valurs of x,y, s. 77= dog and g(x, y,0)=0 where g(x,y)= x + 3y + 4 (8x, 2)) = (2x, 2y+3) x + 3y + y - yoco > 0 axd=8x, n(2y + 3) = 2x = x(4-4)=0 , n(2y+3)=27 - @ =) 2=0, orazy Case when d=4 from © 4(29+3)= 22 - 4y-y= -6 + 3y=-63 y=-2 from 0 x²_674-40=0 > x = 1 162 = I 742 points (1.2), (192,-2) = ( 142,-2), (-/42,-2) Case ② when no from 0 - 3y+y²-40=07 Y+39-40 =0 (+ 8% -5= 3) y = 5, -8 points are cos),(0,8) Now at 1542,-2), fing) = 4x 42 + 4 = 172 at 1-142,-2) ,H1,0)= 4x42+ 4 = 172 at 10,52 fing) = 4x0 + 25= 25 at (0,8), finy)=4 X0 +69=64 a tmua = 1172, fmin= (25]