In Exercises 29-31, suppose the revenue R, in thousands of...

In Exercises 29-31, suppose the revenue $R,$ in thousands of dollars, from producing and selling $x$ hundred LCD TVs is given by $R(x)=-5 x^{3}+35 x^{2}+155 x$ for $0 \leq x \leq 10.07$. Use a graphing utility to graph $y=R(x)$ and determine the number of TVs which should be sold to maximize revenue. What is the maximum revenue?

In Exercises 29-31, suppose the revenue $R,$ in thousands of dollars, from producing and selling $x$ hundred LCD TVs is given by $R(x)=-5 x^{3}+35 x^{2}+155 x$ for $0 \leq x \leq 10.07$.
Use a graphing utility to graph $y=R(x)$ and determine the number of TVs which should be sold to maximize revenue. What is the maximum revenue?

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Key Concepts

Revenue Function

A revenue function is a mathematical model that represents income generated from selling a specific number of units. It encapsulates the relationship between the quantity sold and the total revenue, allowing businesses to analyze performance and make informed decisions about production and pricing strategies.

Optimization

Optimization in mathematics involves finding the maximum or minimum values of a function within a given set of constraints. When applied to revenue functions, it helps determine the most efficient production or sales level that maximizes profit or revenue, ensuring resources are used effectively.

Graphical Analysis

Graphical analysis involves plotting a function to visualize its behavior, including its increasing and decreasing intervals and points where it attains local maxima or minima. This method is particularly useful for understanding complex functions and confirming analytical results in real-world applications.

Domain Analysis

Domain analysis is the process of identifying the set of all possible input values for a function that are meaningful within a real-world context. In applied problems, such as maximizing revenue, the domain is restricted to values that are feasible and practical, ensuring the solutions are relevant to the situation at hand.

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