The amount of revenue a company makes per day by selling x items is given by the function f(x) = 14x - 0.2x2. How many items should be sold if the company wants to maximize their profit?

To find the number of items that should be sold in order to maximize profit, we need to determine the value of x that yields the maximum value of the profit function.

The profit, P(x), can be calculated by subtracting the costs from the revenue. In this case, the cost is not given, so we only consider the revenue:

Revenue = f(x) = 14x - 0.2x^2

To find the maximum profit, we need to find the maximum value of the profit function. This occurs when the slope of the profit function is zero, which means the derivative of the profit function with respect to x is zero.

Let's find the derivative of the profit function:

P'(x) = d/dx (14x - 0.2x^2)
= 14 - 0.4x

Setting P'(x) = 0, we have:

14 - 0.4x = 0

Solving for x, we find:

0.4x = 14
x = 14 / 0.4
x = 35

Therefore, the company should sell 35 items in order to maximize their profit.