A monopolist manufactures and sells two competing products, call them I and II, that cost $42 and $35 per unit, respectively, to produce. The revenue from marketing x units of product I and y units of product II is R (x, y) = 110x + 127y - 0.04xy - 0.1x^2 - 0.2y^2. Find the exact values of x and y that maximize the monopolist's profits.

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Mathematics

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6 February, 16:00

A monopolist manufactures and sells two competing products, call them I and II, that cost $42 and $35 per unit, respectively, to produce. The revenue from marketing x units of product I and y units of product II is R (x, y) = 110x + 127y - 0.04xy - 0.1x^2 - 0.2y^2. Find the exact values of x and y that maximize the monopolist's profits.

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Avah Obrien · Answer 1

F (x, y) = 116x + 133y - 0.04xy - 0.1x^2 - 0.2y^2

For f (x, y) to be maximum, ∂/∂x [f (x, y) ] = 0 and ∂/∂y [f (x, y) ] = 0

=> ∂/∂x [f (x, y) ] = 0 = > 116 - 0.04y - 0.2x = 0 ... (1)

and ∂/∂y [f (x, y) ] = 0 = > 133 - 0.04x - 0.4y = 0 ... (2)

Solving eqns. (1) and (2),

x = 280 and y = 524.