Let $f(x),g(x)$ be polynomials such that their derivatives $f'(x),g'(x)$ have $n$ and $m$ real roots. What is the possible minimal/maximal numbers of real roots for the polynomial $(f(g(x))'$?
My idea was as follows. Since $(f(g(x))'=f'(g(x)) g'(x)$ then minimal numbers of real roots is $m$ and the maximal number of the roots is $m+n.$ An I right?