Polynomials and Location of Real Roots Edit in hopes of clarifying the question:
I think that for every polynomial $f(x)$  having at least one real zero there is an $x$ for which $f(x) = f'(x)$. Am I right?
Original question:
Let $f(x)$ be a polynomial having at least one real zero. Is it possible for $f(x)$ to be equal to its derivative function anywhere for some value(s) of $x$? Well, as far as I am concerned it does.  
 A: For any polynomial $f(x)$ with a real root, there is always a $y$ such that $f'(y) = f(y)$.
Method I
Let's say $f(a) = 0$. Consider the function $g(x) = e^{-x}f(x)$. We have
$$f(a) = 0 \quad\implies\quad g(a) = 0$$
Pick any arbitrary $b > a$, either $g(b) = 0$ or $g(b) \ne 0$.
For the second case,


*

*By IVT, there is a $p \in (a,b)$ such that $g(p) = \frac12 g(b)$.  

*Since $\lim_{x\to\infty} g(x) = 0$, there is a $q > b$ such that $|g(q)| < \frac12 |g(b)|$.

*By IVT, there is a $r \in (b,q)$ such that $g(r) = \frac12 g(b)$.  


In both cases, we get a pair of numbers
$(u,v) = \begin{cases}
(a,b),& 1^{st}\text{ case}\\
(p,r) & 2^{nd}\text{ case}
\end{cases}$
which satisfies $g(u) = g(v)$.
By Rolle's theorem, there is a $w \in (u,v)$ such that $$g'(w) = 0 \quad\iff\quad f(w) = f'(w)$$
Method II
Following is another argument that is hopefully easier to follow.
WOLOG, we will assume the leading coefficient of $f(x)$ is positive. This means
$$\lim_{x\to+\infty} f(x) = \lim_{x\to+\infty} (f(x) - f'(x)) = +\infty$$
Let's say $f(x)$ has one or more real roots. Since $f(x)$ is a polynomial, it has finitely many real roots.
Let $a$ be the largest one.
Notice $f(x) > 0$ for all $x > a$ implies
$\frac{f(a+h) - f(a)}{h} > 0$ for all $h > 0$. This leads to
$$f'(a) = \lim_{h\to 0}\frac{f(a+h)-f(a)}{h} = \lim_{h\to 0+}\frac{f(a+h)-f(a)}{h} \ge 0$$
If $a$ is a double root, then $f'(a) = 0$ and hence $f(a) = f'(a)$ and we are done.
Otherwise, $f'(a) > 0$ and $f(a) - f'(a)$ is negative. Since $f(x) - f'(x)$ is positive for sufficiently large $x$, IVT tells us there is a $b > a$ such that $f(b)-f'(b) = 0$.
