SUMMARY
The function's first derivative is given by f'(x) = (cos²(x)/x) - 1/5. The critical numbers of the function f on the interval (0,10) are determined by finding where f'(x) equals zero or does not exist. The correct number of critical points is 3, as the equation cos²(x) = 5x has three intersection points when graphed. The confusion arises from mistakenly analyzing the second derivative instead of the first derivative to identify critical points.
PREREQUISITES
- Understanding of calculus concepts, specifically derivatives.
- Familiarity with trigonometric functions and their properties.
- Ability to graph functions and analyze intersections.
- Knowledge of critical points and their significance in function analysis.
NEXT STEPS
- Study the method for finding critical points of functions using first derivatives.
- Learn how to graph trigonometric functions and their transformations.
- Explore the implications of critical points in determining function behavior.
- Practice solving equations involving trigonometric identities and their applications.
USEFUL FOR
Students studying calculus, particularly those focusing on derivatives and critical point analysis, as well as educators looking for clarification on common misconceptions in derivative applications.