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The profit function is the heart of any business calculus problem aiming to determine the most financially beneficial outcomes. For the Hot Wheels Rent-A-Car Company, this function expresses its net profit in terms of the number of customers serviced.

In mathematical terms, a profit function, denoted typically by P(x), represents the relationship between the number of goods sold or services rendered (x) and the profit earned. It's usually derived from the revenue function minus the cost function. In our particular case, for the first 50 customers, the profit remains constant at \(12 per customer, resulting in the straightforward calculation:
\[ P(50) = 12 \times 50 = \)600 \].

However, when the number of customers exceeds 50, the profit per additional customer decreases by 6 cents. This real-world scenario is translated into the profit function as:
\[ P(x) = 600 - 0.06(x - 50) \],
for all values of \( x \) greater than 50. This piecewise definition captures the change in profit dynamics beyond the initial 50 customers.

The derivative is a measure of how a function changes as its input changes. It's the cornerstone of calculus and critical in solving optimization problems. In the context of our profit function, we're interested in how the profit changes with each additional customer; essentially, we want to know the rate at which profit increases or decreases.

To find this, we calculate the derivative of the profit function, denoted as P'(x). The derivative tells us the slope of the profit function at any given point. If we imagine the function as a hill, then the slope of that hill shows how steep it is; if the hill is rising up, the business is earning more profit with additional customers, and if it's going down, the opposite is true.

Since the profit decreases by a constant 6 cents for each customer over 50, we have a linear function for \( x > 50 \), and the derivative is simply the constant \( -0.06 \). This constant slope indicates that the profit is decreasing at a steady rate beyond 50 customers.

Calculating the maximum profit involves finding the point where the profit function reaches its peak. This is where derivatives come into play. By setting the derivative equal to zero and solving for x, we can find the number of customers that maximizes the profit. However, in the case of Hot Wheels Rent-A-Car, since the derivative for \( x > 50 \) is a negative constant, the function does not have a 'peak' in that interval.

For linear functions, the maximum or minimum occurs at the boundaries of the domain. Hence, we determine that the maximum profit is achieved when servicing 50 customers because beyond that point, the profit starts to decrease linearly due to the reduction in net profit per customer.

Therefore, the calculation is straightforward for this specific function: the company achieves its maximum profit of $600 when servicing exactly 50 customers. Increasing the customer base beyond this threshold will only decrease the profitability according to the linear relationship established by the profit function.