Finding critical point between two roots

Question

For finding a critical point like A ($f′(x(A))=0$) between x=α and x=β in $f(x)=(x-α)^m(x-β)^n$ I use this formula: $$x(A)=\frac{α.n+β.m}{m+n}$$ but I am interesting to know is there any formula like this to calculate the approximate or exact abscissa of a critical point between two roots of a function like $f(x)=(x-α)^m(x-β)^n\dots(x-γ)^t$ which has more than two roots?

Answer 1

From

$$ f(x)=(x-α)^m(x-β)^n(x-γ)^t $$

deriving we have

$$ f'(x)=m(x-α)^{m-1}(x-β)^n(x-γ)^t+n(x-α)^m(x-β)^{n-1}(x-γ)^t+t (x-α)^m(x-β)^n(x-γ)^{t-1}=0 $$

and

$$ \frac{f'(x)}{f(x)} = \frac{m}{x-\alpha}+\frac{n}{x-\beta}+\frac{t}{x-\gamma}=0 $$

and now we have a second order polynomial... So for $k$ terms the polynomial to solve will be of order $k-1$