You have to solve for x, i.e., you must find the numbers x that satisfy f'(x) = 0
The polynomial function is: f(x) = x(x-4)^3
I know that you have to find the first derivative of the function, which is:
f'(x): 4x^3 - 36x^2 + 96x - 64
f'(x): x^3 - 9x^2 + 24x - 16
We are supposed to set f'(x)=0, so:
x^3 - 9x^2 + 24x - 16 = 0
I would think that you bring the "-16" to the other side:
x^3 - 9x^2 + 24x = 16
Factor out the x:
x(x^2 - 9x + 24) = 16
But I know that 16 is not one of the critical numbers, so I'm not sure of where I went wrong. If anyone could correct me or provide suggestions, that would be very helpful! Thank you in advance!
A better alternative would be to apply the product and chain rules to evaluate the derivative, as the common factors would most likely appear.