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In mathematics, a **profit function** is used to model the total profit gained by a company for producing a specific quantity of goods or services. A profit function depends on various economic factors, such as production cost and revenue. Generally, it is expressed in terms of a variable, often represented as \(x\), which signifies the number of units produced or sold.
For the given problem, the profit function is \(P(x) = -2x^2 + 26x - 44\). Here, \(x\) denotes the number of units (in thousands), and the quadratic nature of the profit function typically implies that the profit will increase up to a certain point and then decrease.

• Coefficient Analysis: The term \(-2x^2\) indicates that the profit function opens downwards. This signifies there is a ceiling to how much profit can be made by continuously increasing production.
• Interpretation of Terms: Each part of the function \(-2x^2\), \(26x\), and \(-44\) has its role—where \(-2x^2\) represents the marginal decreasing profitability beyond certain production, \(26x\) represents linear gains with increased units, and \(-44\) could represent fixed costs.

Ultimately, solving this profit function helps us understand for what range of \(x\) the company operates at a profit, guiding production decisions.

A **quadratic equation** is a second-degree polynomial represented as \(ax^2 + bx + c = 0\). Solving quadratic equations is essential for finding the roots, where the function equals zero. A root of a quadratic equation signifies points where profit transitions from negative to positive or vice versa.
The quadratic formula is a reliable method to solve these equations:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
  This formula calculates the roots of the quadratic equation by plugging in coefficients from the equation: \(a = -2\), \(b = 26\), and \(c = -44\).

The step-by-step solution involves calculating discriminants and ensuring the correct application of plus/minus signs. The roots, \(x = 2\) and \(x = 11\), represent critical points used in evaluating intervals to determine where profits are positive.
The quadratic equation provides a mathematical foundation for determining the behavior of the profit function over different production levels.

**Interval testing** is a method used to determine the sign of a function in the regions between and beyond the roots of a quadratic equation. When a quadratic function changes sign at its roots, interval testing helps us verify within which intervals the function is positive or negative.
Applying this method involves:

• Identifying Intervals: Use the roots, \(x = 2\) and \(x = 11\), to divide the number line into intervals: \((-\infty, 2)\), \((2, 11)\), and \((11, \infty)\).
• Testing Values: Select a test value from each interval to substitute back into the profit function \(P(x) = -2x^2 + 26x - 44\) to check the sign.
  For instance:
  • \(x = 1\) from \((-\infty, 2)\) yields \(P(1) = -20\), which shows profit is negative.
  • \(x = 5\) from \((2, 11)\) yields \(P(5) = 16\), indicating a positive profit.
  • \(x = 12\) from \((11, \infty)\) results in \(P(12) = -32\), identifying a negative profit.

By confirming the sign change, interval testing allows us to state confidently where the company's profit is positive, guiding those crucial production decisions effectively. For this exercise, the company benefits within the interval \((2, 11)\), meaning producing between 2000 and 11000 units results in a profit.