combinations of the /raming principles Suppose we want to maximize f(x,y) = 12x - x2 + 18y - 3y2 - 10, subject to x +

Y :s; 8. x õ!: 0 and y Õ!: O. You should verify the solution has x = 5.25 and y = 2.75.

Now consider the following. (i) Initially drop the constant of - 10. (ii) Notice that ifthe constraint were not present, we would never set x above 6 or y above 3. Doing so lowers the objective function. Sirnilarly, we would never set x below 6 or y below 3. A sIight increase whenever the variables are below the noted targets wiIl increase the objective function. (iii) This insight implies, with the constraint present, we would never set x below 5 (because y would never be set above 3). (iv)

Together, then, we can locate the best choice of x by maxirnizing 12x - x2 + 18(8 -

x) - 3(8 - X)2, subject to the constraint 5 :s; x :s; 6. Try it.

Carefully document the use of the three principles of consistent frarning in this exeecise.

AppendixLO1