Maximizing Profit. In business, profit is the difference...

Maximizing Profit. In business, profit is the difference between revenue and cost; that is, $$ \begin{array}{l} \text { Total profit }=\text { Total revenue }-\text { Total cost, } \\ P(x)=R(x)-C(x) \end{array} $$ where $x$ is the number of units sold. Find the maximum profit and the number of units that must be sold in order to yield the maximum profit for each of the following. $$ R(x)=5 x, C(x)=0.001 x^{2}+1.2 x+60 $$

Maximizing Profit. In business, profit is the difference between revenue and cost; that is,
$$
\begin{array}{l}
\text { Total profit }=\text { Total revenue }-\text { Total cost, } \\
P(x)=R(x)-C(x)
\end{array}
$$
where $x$ is the number of units sold. Find the maximum profit and the number of units that must be sold in order to yield the maximum profit for each of the following.
$$
R(x)=5 x, C(x)=0.001 x^{2}+1.2 x+60
$$

Key Concepts

Vertex Formula

The vertex formula is a key mathematical tool for quadratic optimization. It determines the precise point where a quadratic function reaches its maximum or minimum value, which, when applied to the profit function, reveals the number of units to be sold for maximum profit.

Quadratic Function Optimization

When the profit or cost function is quadratic, optimization techniques are used to find the maximum or minimum value of the function. This usually involves analyzing the parabola formed by the quadratic equation, which in the context of profit maximization helps identify the optimal production level.

Profit Function

This concept refers to the function that calculates net earnings by subtracting total costs from total revenues. It serves as the objective function in a business optimization problem, where finding its maximum value indicates the most profitable level of production.

Revenue Function

The revenue function models the total income generated by selling a certain number of units. It is crucial in business analysis as it directly reflects how changes in sales volume influence total earnings.

Cost Function

The cost function represents the total expenses incurred in producing a given number of units, including both fixed and variable costs. Understanding this function is essential for evaluating profit, since profit is determined by comparing revenue to these costs.

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