The total daily profit (in dollars) realized by Weston Publishing in publishing

Question

The total daily profit (in dollars) realized by Weston Publishing in publishing and selling its dictionaries is given by the profit function
p(x, y) = 0.005x2 0.003y2 0.002xy + 14x + 12y 200
where x stands for the number of deluxe editions and y denotes the number of standard editions sold daily.

Weston's management has decided that the publication of these dictionaries should be restricted to a total of exactly 400 copies/day. How many deluxe copies and how many standard copies should be published each day to maximize Weston's daily profit? (Give your answers correct to the nearest fully copy.)

____ deluxe copies

____ standard copies

Explanation / Answer

Maximize p(x,y) subject to x+y=400
p(x,y) = -0.005x^2 - 0.003y^2 - 0.002xy + 14x + 12y - 200
f(x) = -0.005x^2 -0.003(400-x)^2 -0.002x(400-x) + 14x + 12(400-x) -200

The variables are in terms of x
df/dx = (-0.005)(2x) + 0.006(400-x) -(0.002)(400) +0.004x + 14 -12 = 0
-0.010x + 2.4 -0.006x -0.8 +0.004x + 2 = 0
x=300

300 copies of deluxe editions and 100 copies of standard editions

d^f/dx^2 = -0.010 -0.006 = -0.016 < 0, so f has been maximized.

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