Find the maximum profit and the number of units that must be...

Find the maximum profit and the number of units that must be produced and sold in order to yield the maximum profit. Assume that revenue, $R(x),$ and $\cos t, C(x)$ are in dollars.
$$R(x)=2 x, \quad C(x)=0.01 x^{2}+0.6 x+30$$

Key Concepts

Quadratic Optimization

When the profit function is quadratic, as in this case, it takes the form of a parabolic equation. Optimizing such a function involves finding its vertex, which represents the maximum (or minimum) profit. Techniques such as using the derivative to set it equal to zero or applying the vertex formula (-b/2a) are standard methods for determining the optimal production quantity.

Profit Function

The profit function is obtained by subtracting the cost function from the revenue function. It indicates the net earning or loss made from producing and selling units. Analyzing the profit function is crucial for making decisions about production levels to maximize profitability.

Cost Function

The cost function represents the total costs incurred in producing a certain number of units. It usually includes fixed costs (independent of the number of units produced) and variable costs (which change with production volume). Understanding the cost function is essential to determining the profitability of a production process.

Revenue Function

The revenue function represents the total income generated from selling a certain number of units, typically calculated as the product of the number of units sold and the price per unit. It provides a basis for understanding how sales volume impacts overall income and is a fundamental component in analyzing business performance.