Effect on roots of function on taking the derivative of the function

Question

Suppose there is a function $$f(x)=(x-1)^{15}(x-2)^{20}(x-3)^{25}(x-4)^{30}$$ As we take the derivatives of the function, what will happen to the number of real roots and the number of distinct real roots? Can we explain it with the help of IMVT or Rolle's theorem?

Answer 1

If $(x-r)^n|P(x)$, then $(x-r)^{n-1}|P'(x)$. Then using Rolle's should give you a pretty good idea.