Question

The company's annual profit (in thousands of dollars) as a function of the price of a bracelet (in dollars) is modeled by: P(x)=−2x2+16x−24P(x)=-2x^2+16x-24 P(x)=−2x 2 +16x−24

Answer 1

The maximum profit that the company can ear = $8,000

Explanation:

The company's annual profits is given through the relation

P(x) = -2x² + 16x - 24

where P is in thousand of dollars and x is the price of one bracelet in dollars.

This problem requires us to obtain the maximum of the function that represents the maximum profit the firm can earn.

At maximum profit, (dP/dx) = 0 and (d²P/dx²) < 0.

P(x) = -2x² + 16x - 24

(dP/dx) = -4x + 16

At maximum point,

(dP/dx) = -4x + 16 = 0

x = 4

(d²P/dx²) = -4 < 0 (the value of x really does correspond to the maximum point of the Profit function)

So, to obtain the maximum profit, we substitute the value of x, the price, that corresponds to the maximum of that function into the Profit's function.

P(x) = -2x² + 16x - 24

P(max) = -2(4²) + 16(4) - 24 = 8 thousand dollars

Hope this Helps!!!