Total Profit. Derex, Inc., determines that its total profit...

Total Profit. Derex, Inc., determines that its total profit function is given by
$$P(x)=-3 x^{2}+630 x-6000.$$
a) Find all values of $x$ for which Derex makes a profit.
b) Find all values of $x$ for which Derex loses money.

Key Concepts

Profit and Loss Conditions

The conditions under which a business makes a profit or incurs a loss are determined by the sign of the profit function. When the profit function is greater than zero, the business is profitable; when it is less than zero, the business experiences a loss. This concept is crucial for evaluating and managing financial performance.

Quadratic Function

A quadratic function is a polynomial described by an equation in the form ax² + bx + c. Its graph is a parabola, and its key features include its vertex, axis of symmetry, and the roots where the function changes sign, which are essential for analyzing real-world scenarios such as profit and loss.

Profit Function

A profit function represents the difference between total revenue and total costs as a function of the level of activity or units sold. It is used to determine the levels of production or sales that yield a profit and to identify break-even points where profit transitions to loss.

Quadratic Inequalities

Quadratic inequalities involve determining the intervals on the x-axis where a quadratic expression is positive or negative. This requires solving the quadratic equation to find its roots and then using these critical points to establish the intervals of profit (positive values) or loss (negative values).

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