A manufacturer sells video games with the following cost and revenue functions (

Question

A manufacturer sells video games with the following cost and revenue functions (in dollars), where x is the number of games sold. Determine the interval(s) on which the profit function is increasing Co-0.172-0.000082 R(x)-0.746x2-0.0002x3 Select the correct choice below and fill in any answer boxes within your choice. O A. The profit function is increasing on the interval(s) (Type your answer in interval notation. Use a comma to separate answers as needed.) The profit function is never increasing. O B. Click to select and enter your answerls) O corch or enter website name

Explanation / Answer

Profit function,
P(x) = R(x) - C(x)
= (0.746*x^2 - 0.0002*x^3) - (0.17*x^2 - 0.00008*x^3)
= 0.576*x^2 - 0.00012*x^3

differentiate with respect to x:
P’(x) = d/dx (0.576*x^2 - 0.00012*x^3)
= 0.576*2*x - 0.00012*3*x^2
= 1.152*x - 0.00036*x^2

Put P’(x) = 0
1.152*x - 0.00036*x^2 = 0
x (1.152 - 0.00036*x) = 0
x = 0 and x = 1.152/0.00036 = 3200

P’’(x) = d/dx(1.152*x - 0.00036*x^2)
= 1.152 - 0.00072*x
at x=3200
P’’(3200) = 1.152 - 0.00072*x
= 1.152 - 0.00072*3200
= -1.152

Since P’’(3200) is negative, x = 3200 is maxima

At x< 3200, P’(x) is positive
So, P(x) is increasing for x< 3200

Answer: (0,3200)

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