Final answer:
To find the partition numbers of f'(x), we need to calculate the derivative of f(x), set it equal to zero, and solve for x. These partition numbers are critical points where the first derivative is either maximum, minimum or an inflection point.
Step-by-step explanation:
To find all partition numbers of f'(x), which is the derivative of the function f(x) = x³ - 8x² - 12x + 4, we first need to calculate the derivative.
The derivative of the function is:
f'(x) = 3x² - 16x - 12.
Partition numbers are essentially the critical points of the function, and they occur where the first derivative is equal to zero. So, we solve for x in the equation:
3x² - 16x - 12 = 0.
Factoring or using the quadratic formula would give us the roots of this equation, which are the partition numbers.