SUMMARY
The critical numbers of the function f(x) = x*sqrt(2x + 1) are -1/3 and -1/2. The derivative f'(x) is calculated as (3x + 1)/sqrt(2x + 1). Setting the numerator equal to zero, 3x + 1 = 0, confirms that x = -1/3 is a critical number. The discussion also identifies -1/2 as another critical number, providing a complete analysis of the function's critical points.
PREREQUISITES
- Understanding of calculus concepts, specifically derivatives
- Familiarity with critical numbers and their significance in function analysis
- Knowledge of square root functions and their properties
- Ability to solve algebraic equations
NEXT STEPS
- Study the application of the First Derivative Test for identifying local extrema
- Learn about the implications of critical numbers on the graph of a function
- Explore the concept of concavity and inflection points in relation to critical numbers
- Investigate the use of graphing calculators or software for visualizing functions and their derivatives
USEFUL FOR
Students studying calculus, mathematics educators, and anyone interested in understanding the behavior of functions through critical number analysis.