If the profit function for a product is P(x) = 2400x + 30x2 − x3 − 26,000 dollars, selling how many items, x, will produce a maximum profit? x = items Find the maximum profit.

Selling 40 items will produce a maximum profit.

Step-by-step explanation:

We need to use First and Second Derivative Tests on profit function to determine how many items will lead to maximum profit. Let , where is the profit for a product, measured in US dollars, and is the amount of items, dimensionless.

First we derive the profit function and equalize it to zero:

(Eq. 1)

Roots are found by Quadratic Formula:

and

Only the first root may offer a realistic solution. The second derivative of the profit function is found and evaluated at first root. That is:

(Eq. 2)

(Absolute maximum)

Therefore, selling 40 items will produce a maximum profit.

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