A soft-drink vendor at a popular beach analyzes his sales...

A soft-drink vendor at a popular beach analyzes his sales records and finds that if he sells $x$ cans of soda pop in one day, his profit (in dollars) is given by
$$
P(x)=-0.001 x^2+3 x-1800
$$
What is his maximum profit per day, and how many cans must he sell for maximum profit?

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Key Concepts

Quadratic Functions

Quadratic functions are polynomial functions of degree two that take the form ax² + bx + c. They are characterized by their parabolic graphs, which can open upward or downward depending on the sign of the coefficient a. This concept is fundamental in algebra and calculus because it models various real-world phenomena, including projectile motion and profit maximization scenarios.

Vertex of a Parabolic Graph

The vertex of a parabola is the point where the function attains its extreme value, either a maximum or a minimum. In the context of quadratic functions, the vertex can be computed using the formula x = -b/(2a), which provides the x-coordinate of the vertex. The corresponding y-value is found by substituting this x back into the function. This concept is crucial for understanding and finding optimum values in models represented by quadratic functions.

Optimization

Optimization in mathematics involves finding the maximum or minimum of a function under given conditions. When a problem is modeled by a quadratic function, the vertex represents the optimal solution—maximum in the case of a downward-opening parabola and minimum in an upward-opening one. This principle is widely applied in economics, engineering, and other fields to maximize performance or profit and minimize errors or costs.

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