Find the maximum value of fx y xy subject to x 2y 44 fmax

Find the maximum value of

f(x, y) = xy

subject to

x + 2y = 44.

f_max

=  

Also find the corresponding point

(x, y).

(x, y) =

Solution

f(x, y) = xy

GIVEN x + 2y = 44

=>x=44-2y

f(y) = (44-2y)y

f(y) = 44y-2y²

for maximum df/dy=0 ,d²f/dy²<0

df/dy =44-4y =0

=>y =11

d²f/dy²=-4<0

when y =11

x=44-2*11

x=22

fmax =xy

fmax =22*11

fmax=242

occurs at point (x,y)=(22,11)