The demand for a fitness video game roughly follow the equation where p is the u

Question

The demand for a fitness video game roughly follow the equation

where p is the unit price and is the number of units sold. It costs the supplier

dollars to sell a total of units.

(a) Write profit as a function of (assuming that the level of demand drives production). (b) At what production level is profit maximized? What is the maximum profit? (c) At what production level is profit decreasing most rapidly?

Explanation / Answer

a) profit function: f = p*x - C = 140x - 4x^1.5 - (-0.02x^2 + 35x + 500) = 0.02x^2 - 4x^1.5 + 105x - 500 b) for max profit: df/dx = 0 and d2f / dx^2 < 0 => df/dx = f ' = 0.04x - 6x^0.5 + 105 = 0 This is quadratic in sqrt(x) on solving it,we get: root x = 129.77 or 20.23 or x=409.25 or 16840.25 f (409.25) = 12704.56 f(16840.25) = -ve f(409) =12704.56 > f(410) b) For max. profit production level should be x = 409.25 (If x is an integer then it should be 409 units) The maximum profit is 12704.56 c) Here we want x, at which f ' is minimum => f '' = 0.04 - 3x^(-0.5) = 0 => x = (3/0.04)^2 = 5625 (This is out of range of x) So, it should be corner point. So with increasing x , the profit will be decreasing most rapidly. x= as large as possible for production.

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