Final answer:
To find the critical points of the function f(x) = x⁵ - 10x³ - 8, we need to find the values of x where the derivative of f(x) equals zero or is undefined. The critical points are the values of x that make the derivative equal to zero.
Step-by-step explanation:
To find the critical points of the function f(x) = x⁵ - 10x³ - 8, we need to find the values of x where the derivative of f(x) equals zero or is undefined. The critical points are the values of x that make the derivative equal to zero.
First, find the derivative of f(x) and set it equal to zero: f'(x) = 5x⁴ - 30x² = 0. Factor out x² to get x²(5x² - 30) = 0. Set each factor equal to zero: x² = 0 and 5x² - 30 = 0. Solving these equations, we find that x = 0 and x = ±√6 are the critical points of the function f(x).