SUMMARY
The problem involves finding the value of b in the cubic function y = x^3 + ax^2 + bx - 8, given that there is a point of inflection at (2,0). The second derivative, y'' = 6x + 2a, is set to zero at x = 2, leading to the conclusion that a = -6. Substituting this value into the first derivative y' = 3x^2 + 2ax + b allows for further analysis. Ultimately, the value of b is determined to be 8.
PREREQUISITES
- Understanding of cubic functions and their derivatives
- Knowledge of points of inflection in calculus
- Ability to solve equations involving derivatives
- Familiarity with polynomial functions
NEXT STEPS
- Study the concept of points of inflection in greater detail
- Learn how to apply the second derivative test for concavity
- Explore the implications of cubic function behavior on graphing
- Investigate the relationship between coefficients and function shape in polynomials
USEFUL FOR
Students studying calculus, particularly those focusing on derivatives and points of inflection, as well as educators seeking to explain these concepts effectively.