- The critical points of the inequality \(x^2 - 8x + 15 < 0\) are the roots of the equation \(x^2 - 8x + 15 = 0\).
- Factoring the quadratic gives \((x - 3)(x - 5) = 0\).
- Solving for \(x\) yields the roots \(x = 3\) and \(x = 5\).
- Comparing with the given options, the correct critical points are \(\boxed{3, 5}\).
Explanation
- Analysis of the Problem
To solve the inequality \(x^2 - 8x + 15 < 0\), we first identify the critical points. These points are defined as the roots of the corresponding quadratic equation \(x^2 - 8x + 15 = 0\). The critical points divide the number line into distinct intervals, which helps us determine where the quadratic expression is negative.
- Finding the Critical Points
We solve the quadratic equation \(x^2 - 8x + 15 = 0\) by factoring. We look for two numbers whose product is \(15\) and whose sum is \(-8\). These numbers are \(-3\) and \(-5\). Therefore, the equation can be factored as:
\[(x - 3)(x - 5) = 0\]
Setting each factor equal to zero, we find the roots:
- \(x - 3 = 0 \implies x = 3\)
- \(x - 5 = 0 \implies x = 5\)
Thus, the critical points for the inequality are \(3\) and \(5\).
- Comparing with Given Options and Conclusion
We compare the calculated critical points \(x = 3\) and \(x = 5\) with the given options: \(-3, -15, 5, 3, -5\). The values that match our results are \(3\) and \(5\). Since the inequality is strictly less than zero (\(x^2 - 8x + 15 < 0\)), the solution interval is the open interval \((3, 5)\). The points that define this interval, known as the critical points, are \(3\) and \(5\). Therefore, the correct choices are \(3\) and \(5\).
Examples
In engineering, critical points are often used to identify the range of operating conditions where a system remains stable. For example, if the performance of a mechanical component is modeled by a quadratic expression, finding the roots helps determine the "safe" range of values for an input parameter (like speed or load) to ensure the value stays below a specific limit (e.g., temperature or stress). By calculating these points, engineers can design systems that operate reliably within the required bounds.