If $f$ is a polynomial in one variable with real coefficients which has all its roots real, then its derivative $f'$ has all its roots real Is  the following statement  true/false ?

If  $f$ is  a polynomial in one variable  with real coefficients which  has  all its roots real, then  its derivative  $f'$ has all its roots real as  well 

My attempt  : I think  this  statement is false. Take $f(x) =  \frac{1}{3} x^3 + x$ and now $f'(x) = x^2 + 1  $ , $x^2+ 1=0 $ implies  $x= i,-i$  which does not belong to $\mathbb{R}$,   so given the above question statement is false
Edits  :another  counter example   $f(x) = x+1$ , but $f'(x) =1$   has  no root in $\mathbb{R}$
Is  its  true ?
 A: Note that the only irreducible polynomials in $\mathbb{R}$ are quadratic and linear, that is, every polynomial can be broken down into a product of linear or quadratic polynomials. So it is enough to analyze the problem for the case of polynomials of grade 1 and 2. It is easy to see that when the polynomial has grade 1 there is nothing to do. Let's analyze the case of grade 2.
Let $f(x) = ax^{2} +bx+c$ so $f^{\prime}(x) = 2ax +b$. Now, if $f^{\prime}(x)$ have a complex root then $x=- \frac{b}{2a} \in \mathbb{C}$ which implity that $x= - \frac{b}{2a} \pm \frac{\sqrt{b^{2}-4ac} }{2a} \in \mathbb{C}$, i. e., $f(x)$ have a complex root. 
A: Whether f'(X) will have real roots or not depend on the number of times the differentiable function changes its direction FROM INCREASING TO DECREASING OR FROM DECREASING TO INCREASING.
$$let\,\,f(x)=x^3-x $$
$$f(x)=0 at x=0,1,-1 $$
f(x) is continuous hence twice will change its direction to become zero once between -1 and 0 and another time between 0 and 1.
$$f'(x)=3x^2-1=0\,\,at \,\,x=\pm\frac{1}{\sqrt{3}}$$
Whereas if you consider a function monotonically increasing or decreasing e.g $f(x)=x^3+x + 1 $ is increasing hence even though f(x) = 0 for one of the real value of x but f'(x) will have complex roots only.
With exception as in case of function of form f(x)=$x^n$ for $n\in N$. 
A: The roots of f’(x) always lie in the convex hull of the roots of f(x). Hence if f(x) has only real roots, the convex hull is a subset of the real line which in turn implies that the roots of f’(x) are also real. 
