The weekly demand for the Pulsar 40 -in. high-definition...

The weekly demand for the Pulsar 40 -in. high-definition television is given by the demand equation $p=-0.05 x+600 \quad(0 \leq x \leq 12,000)$ where $p$ denotes the wholesale unit price in dollars and $x$ denotes the quantity demanded. The weekly total cost function associated with manufacturing these sets is given by $$ \begin{array}{l} \text { If } C(x)=0.000002 x^{3}-0.03 x^{2}+400 x+80,000 \\ \text { ( } x=0,00002 x^{3}-0,00 x+0000=00000000000000000000000000000000000000 \\ \text { (?) } 6,0000 \end{array} $$ where $C(x)$ denotes the total cost incurred in producing $x$ sets. Find the level of production that will yield a maximum profit for the manufacturer. Hint: Use the quadratic formula.

The weekly demand for the Pulsar 40 -in. high-definition television is given by the demand equation
$p=-0.05 x+600 \quad(0 \leq x \leq 12,000)$ where $p$ denotes the wholesale unit price in dollars and $x$ denotes the quantity demanded. The weekly total cost function associated with manufacturing these sets is given by
$$
\begin{array}{l}
\text { If } C(x)=0.000002 x^{3}-0.03 x^{2}+400 x+80,000 \\
\text { ( } x=0,00002 x^{3}-0,00 x+0000=00000000000000000000000000000000000000 \\
\text { (?) } 6,0000
\end{array}
$$
where $C(x)$ denotes the total cost incurred in producing $x$ sets. Find the level of production that will yield a maximum profit for the manufacturer.
Hint: Use the quadratic formula.

Key Concepts

Optimization in Business Calculus

Optimization involves finding the maximum or minimum of a function, which is fundamental in business calculus for making decisions that maximize profit or minimize cost. It typically involves setting the derivative of the profit function equal to zero to locate critical points and then analyzing these points to determine the optimum production level.

Profit Function

The profit function is derived by subtracting the total cost function from the total revenue function. It is a key concept in business optimization problems, as it helps in assessing the profitability of production levels and informs decisions aimed at maximizing profit.

Quadratic Formula

The quadratic formula is an algebraic method used to solve quadratic equations of the form ax^2 + bx + c = 0. In the context of profit maximization, when the profit function simplifies to a quadratic form or its derivative does, the quadratic formula provides a systematic way to identify the critical points where maximum profit may occur.

Revenue Function

The revenue function represents the total income a firm receives from selling its product, calculated by multiplying the price per unit by the quantity sold. It is essential in profit analysis because it directly links market behavior, dictated by the demand function, with the income generated from sales.

Demand Function

The demand function defines the relationship between the price of a product and the quantity demanded by consumers. It is a fundamental concept in economics that captures how changes in price affect market demand, and it is used to derive other functions like revenue when combined with the quantity sold.

Cost Function

The cost function describes the total cost incurred in the production of goods, accounting for fixed and variable costs. This function is crucial in business calculus as it provides a basis for determining how different levels of production affect overall expenditures.

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