Second derivative of a third degree polynomial function Let $f(x) = ax^3 + bx^2 + cx +d $ be a third degree polynomial function.
$$f'' = 6ax + 2b = 0 \Longrightarrow x = \frac{-b}{3a} $$
This is equal to $\frac{1}{3}$ of the sum of the roots of $f(x)$. So my question is: can we say that the root of the second derivative is equal to $\frac{1}{3}$ of the sum (of roots) for all third degree polynomial functions? If not, why?

If the zero of $f''(x)$ is $p$. then the polynomial$f(x)$ has root $p$ with multiplicity 3. i.e,$(x-p)^3|f(x)$ and $(x-p)^4$ does not divides$ f(x)$
$(x-p)^3|f(x)$ and $f(x)$ is cubic polynomial $\implies$ $f(x)=k(x-p)^3$ , where $k$ is a real number.
therefore roots of $f(x)$ are $p,p,p$.........................(1)
Sum of the roots$=\frac{-b}{a} = p+p+p=3p$ $\implies$
  $\frac{1}{3}$( Sum of the roots)$=p$...................(2) 
from (1) and (2) , $\frac{1}{3}$( Sum of the roots) is also a root of $f(x)$.
this property is not available for all cubic equations.
