The problem we have on hand is:
To solve the problem, we will solve for the extreme values of xy, and choose from the extreme values the largest value of |xy|.
To do this, we would use Lagrange multipliers:
Set up equations by equating the partial derivatives with respect to x, y and L (the Lagrange multiplier) to zero.0
Zx(x,y)=y+L(86xy-6y+4x+48) = 0
Zy(x,y)=x+L(4y+43x^2-6x+5) = 0
We now need to solve the algebraic system for x and y by first eliminating L from the first two equations to get the following:
f(x)=2y^2+43x^2y-6xy+5y+2x^2+48x-174.3 = 0
we obtain an algebraic system from which we need to extract the real roots.
This can be obtained by using Newton's method (in two variables) with starting points (-0.8,4) and (0.6,5) by substituting trial integer values of y from 1-5.
The solutions are
(-0.8491, 4.2984) giving xy=-3.6500
(0.6472, 4.9170) giving xy=3.1821
So the largest possible value of |xy|=3.64997 (approximately)
It is possible that the solution be obtained analytically (not numerically) using some skilled manipulations.