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March 28, 2017

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The real numbers x and y satisfy the nonlinear system of equations
{2x2−6xy+2y2+43x+43yx2+y2+5x+5y==174,30.
Find the largest possible value of |xy

  • maths - ,

    The problem we have on hand is:
    Maximize |xy|
    subject to:
    f(x,y)=0
    where
    f(x,y)=2y^2+43x^2y-6xy+5y+2x^2+48x-174.3

    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:

    Objective function,
    Z(x,y)=xy+L(f(x,y))
    constraint:
    f(x,y)

    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
    ZL(x,y)=2y^2+43x^2*y-6xy+5*y+2x^2+48x-174.3=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:
    g(x,y)=x(86xy-6y+4x+48)-y(4y+43x^2-6x+5)=0
    Together with
    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
    and
    (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.

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