Final answer:
The critical numbers of the function f(x) = x³ - 3x² - 45x are found by setting its derivative, f'(x) = 3x² - 6x - 45, equal to zero and solving the resulting equation, which yields critical numbers at x = 5 and x = -3.
Step-by-step explanation:
To find the critical numbers of the function f(x) = x³ - 3x² - 45x, we must first find f'(x), the derivative of the function. The critical numbers occur where the derivative is zero or undefined, which corresponds to possible maxima, minima, or points of inflection on the graph of the function.
The derivative of f(x) is calculated as follows:
To find where the derivative is zero, we set f'(x) = 0 and solve for x:
- 3x² - 6x - 45 = 0
- Divide through by 3: x² - 2x - 15 = 0
- Factor: (x - 5)(x + 3) = 0
- Solving this gives us x = 5 and x = -3
Therefore, the critical numbers of the function f(x) are x = 5 and x = -3.