2x + y = 12 (given). ====>
y = 12-2x
xy = x*(12-2x) = -2x^2 + 12x
This is an equation of the quadratic function (parabola). Since it has negative leading coefficient at x^2,
this parabola is turned downward and has a maximum.
For a general equation of a quadratic function q(x) = ax^2 + bx + c with a < 0,
it has a maximum at x = .
In your case a= -2, b= 12 and c= 0, so your parabola achieves its maximum at x= = 3.
And the value of the maximum is the value of this quadratic function at x= 3:
= -18 + 36 = 18.
Answer. The largest possible value of the product xy at given condition is 18.
