A manufacturer sells video games with the following cost and...

A manufacturer sells video games with the following cost and revenue functions (in dollars), where $x$ is the number of games sold, for $0 \leq x \leq 3300$ . $$\begin{array}{l}{C(x)=0.32 x^{2}-0.00004 x^{3}} \\ {R(x)=0.848 x^{2}-0.0002 x^{3}}\end{array}$$ Determine the interval(s) on which the profit function is increasing.

A manufacturer sells video games with the following cost and revenue functions (in dollars), where $x$ is the number of games sold, for $0 \leq x \leq 3300$ .
$$\begin{array}{l}{C(x)=0.32 x^{2}-0.00004 x^{3}} \\ {R(x)=0.848 x^{2}-0.0002 x^{3}}\end{array}$$
Determine the interval(s) on which the profit function is increasing.

Key Concepts

Profit Function

The profit function is defined as the difference between the revenue function and the cost function. It quantifies the net gain from selling goods or services, and analyzing its behavior (such as determining when it is increasing) is crucial for making informed business and production decisions.

Revenue Function

The revenue function represents the total income generated from selling a product or service, often expressed as a function of the quantity sold. It is fundamental for calculating profit and assessing the performance of business operations as sales volume changes.

Cost Function

The cost function represents the total expenditure incurred in the production of goods or services, typically depending on the production quantity. It is essential in profit analysis because it must be subtracted from the revenue to determine the net profit.

Derivative

The derivative of a function measures the rate at which the function’s value changes with respect to changes in its input. In the context of a profit function, the derivative indicates whether profit is increasing or decreasing at a particular level of sales.

First Derivative Test

The first derivative test is a method used in calculus to determine the intervals on which a function is increasing or decreasing by analyzing the sign of its derivative. If the derivative is positive over an interval, it indicates that the function is increasing on that interval, which is key for understanding profit behavior.

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