A manufacturer finds that the revenue generated by selling x...

A manufacturer finds that the revenue generated by selling $x$ units of a certain commodity is given by the function $R(x)=80 x-0.4 x^2$, where the revenue $R(x)$ is measured in dollars. What is the maximum revenue, and how many units should be manufactured to obtain this maximum?

Key Concepts

Revenue Functions

Revenue functions model the income generated from the sale of goods or services and are often expressed as mathematical functions. Analyzing these functions, particularly when they are quadratic, helps in determining the production levels that yield maximum revenue, making them an important tool in business and economics.

Optimization

Optimization involves finding the conditions that maximize or minimize a particular function. In many real-world scenarios, such as financial or operational planning, identifying the maximum revenue or minimum cost is vital. When the function is quadratic, the process typically includes finding the vertex of the parabola.

Vertex of a Parabolic Graph

The vertex of a parabola is the point where the function reaches its maximum or minimum value. For a quadratic function, the vertex can be found using the formula -b/(2a) to calculate the x-coordinate, with the corresponding y-coordinate computed by substituting back into the function. This concept is crucial in optimization problems where identifying the optimal point is necessary.

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically written in the form ax² + bx + c. It forms a parabolic graph that either opens upward or downward depending on the sign of the coefficient a. Understanding quadratic functions is essential for solving optimization problems because their graphical properties, like symmetry and the vertex, allow determination of maximum or minimum values.

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