Using Lagrange multipliers, we have the Lagrangian

with partial derivatives (set equal to 0)




Substituting the first three equations into the fourth allows us to solve for

:

For each possible value of

, we get two corresponding critical points at

.
At these points, respectively, we get a maximum value of

and a minimum value of

.
(6, 2 )
substitute x = 5 into the equation and solve for y
6y + 8 = 20 ( subtract 8 from both sides )
6y = 12 ( divide both sides by 6 )
y =
= 2
the ordered pair is (5, 2 )
50:
40 is 10 less than 50
60 is 10 more than 50
104 is the answer to your problem
860000 2 quart containers of syrup can be made. Good luck on your schoolwork!