The total profit P(x) for a company producing x thousand...

The total profit $P(x)$ for a company producing $x$ thousand units is given by the function $P(x)=-2 x^{2}+26 x-44$. Find the values of $x$ for which the company makes a profit. [Hint: The company makes a profit when $P(x)>0$.]

Key Concepts

Quadratic Functions

A quadratic function is a second-degree polynomial of the form ax² + bx + c, where a, b, and c are constants and a is not equal to zero. These functions graph as parabolas, and their properties—such as the vertex, axis of symmetry, and the roots—are essential in analyzing the behavior of the function, including determining intervals where the function is positive or negative.

Solving Quadratic Inequalities

Solving quadratic inequalities involves finding the roots of the quadratic equation (where the expression equals zero) and then determining the intervals over which the quadratic expression is positive or negative. This typically requires using methods such as factoring or the quadratic formula, and analyzing the sign changes of the quadratic function around the critical points. Understanding these intervals helps in making decisions based on the function's behavior, such as profit analysis in business contexts.

Profit Function Analysis

A profit function represents the profit earned as a function of the number of units produced or sold. When represented as a quadratic function, the analysis of its inequality (i.e., when the profit is greater than zero) is crucial to determine the production levels that yield a positive profit. This analysis links algebraic methods with practical business decision-making by identifying the range of production values that ensure profitability.

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