The total-cost and total-revenue functions for producing x...

The total-cost and total-revenue functions for producing $x$ items are
$$
C(x)=5000+600 x \text { and } R(x)=-\frac{1}{2} x^{2}+1000 x
$$
where $0 \leq x \leq 600 .$
a) Find the total-profit function $P(x)$
b) Find the number of items, $x,$ for which the total profit is a maximum.

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Key Concepts

Profit Function

The profit function is defined as the difference between the revenue and the cost functions. It is used to determine how much profit is made at different levels of production or sales and forms the basis for analyzing financial performance in economics and business.

Revenue and Cost Functions

Revenue and cost functions represent, respectively, the income generated from sales and the expenses incurred in production. They are crucial in financial modeling as they allow the formulation of profit through the relationship P(x) = R(x) - C(x), providing insights into the overall feasibility and efficiency of business operations.

Quadratic Functions

Quadratic functions, which take the form ax² + bx + c, often arise in profit and revenue analysis due to the inclusion of squared terms and are characterized by a parabolic graph. Understanding their properties, such as the vertex, helps in determining the maximum or minimum values crucial for optimization.

Optimization in Economics

Optimization involves finding the best outcome under given constraints, such as maximizing profit or minimizing cost. In problems modeled by quadratic functions, optimization is typically achieved by locating the vertex of the parabola, often using the formula x = -b/(2a), which provides the production level yielding maximum profit.

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