Question

The total profit Upper P (x )​(in thousands of​ dollars) from the sale of x hundred thousand pillows is approximated by Upper P (x )equals negative x cubed plus 15 x squared plus 72 x minus 200​, x greater than or equals 5. Find the number of hundred thousands of pillows that must be sold to maximize profit. Find the maximum profit.

Answer 1

x=12 maximizes the profit function.

Explanation:

We are given that the profit is
`P(x) = -x^3+15x^2+72x-200` for
`x\geq 5`

To find x that maximizes P, we will find the derivative of P(x) and find x such that P'(x) =0. Recall that the derivative of a function of the form
`x^k` is
`kx^(k-1)`, and that the derivative of a constant is zero. Then, by using the properties of derivatives, we get (the details of the calculation is omitted).

`P'(x) =-3x^2+30x+72`.

We want to solve
`P'(x) = 0`. By dividing the equation by -3, we get

`x^2-10x-24=0=(x-12)(x+2)`

So we have that x=12 and x=-2 are solutions. In this case, we are only considering x greater than o equals 0. So, we take x=12.

We will check that x=12 is a maximum of P.

To do so, we will use the second derivative criteria, which is as follows. Given a function f whose first and second derivative exist, a point x is a maximum if
`f''(x)<0`. In our case,

`P''(x) = -6x+30`

Note that
`P''(12) = -42<0`. So x=12 is a maximum of P.