SUMMARY
The critical number of the function f(x) = x³ + 3x² - 24x is determined by finding where its derivative, f'(x) = 3x² + 6x - 24, equals zero. The derivative simplifies to 3(x + 2)(x - 4), yielding critical points at x = -2 and x = 4. Understanding the quadratic nature of the derivative is essential for solving for critical numbers. Familiarity with factoring and the quadratic formula is crucial for accurately identifying these points.
PREREQUISITES
- Understanding of calculus concepts, specifically critical numbers
- Knowledge of derivatives and their significance in function analysis
- Familiarity with quadratic equations and factoring techniques
- Ability to apply the quadratic formula for solving equations
NEXT STEPS
- Study the quadratic formula and its applications in calculus
- Practice solving for critical numbers in various polynomial functions
- Explore the relationship between critical numbers and maximum/minimum values
- Review the process of finding and interpreting derivatives in calculus
USEFUL FOR
Students studying calculus, particularly those focusing on critical numbers and optimization, as well as educators teaching polynomial functions and their derivatives.
