Respuesta :
Answer:
10 hundred thousands of pillows must be sold to maximize the profit of 900 thousands dollars.
Step-by-step explanation:
We are given the following in the question:
[tex]P(x) = -x^3 + 9x^2 + 120x - 200\\x\geq 5[/tex]
Where P(x) is the profit in thousand of dollars and x is the hundred thousand pillows sold.
First, we differentiate P(x) with respect to x, to get,
[tex]\dfrac{d(P(x))}{dx} = \dfrac{d(-x^3 + 9x^2 + 120x - 200)}{dx} = -3x^2 + 18x+120[/tex]
Equating the first derivative to zero, we get,
[tex]\dfrac{d(P(x))}{dx} = 0\\\\-3x^2 + 18x+120 = 0[/tex]
Solving, with the help of quadratic formula, we get,
[tex]x =-4, x = 10[/tex]
Again differentiation P(x), with respect to x, we get,
[tex]\dfrac{d^2(P(x))}{dx^2} = -6x + 18[/tex]
At x = 10
[tex]\dfrac{d^2(P(x))}{dx^2} < 0[/tex]
Thus,by double differentiation test maxima occurs at x = 10 for P(x).
Thus, 10 hundred thousands of pillows that must be sold to maximize profit.
Maximum Profit
[tex]P(10) = -(10)^3 + 9(10)^2 + 120(10) - 200 = 900[/tex]
Thus, the maximum profit is 900 thousands of dollars.
