Putnam 1956 A4

Suppose that the $n$ times differentiable real function $f(x)$ has at least $n+1$ distinct zeros in the closed interval $[a,b]$ and that the polynomial $P(z)=z^n +c_{n-1}z^{n-1}+\ldots+c_1 x +c_0$ has only real zeroes. Show that $f^{(n)}(x)+ c_{n-1} f^{(n-1)}(x) +\ldots +c_1 f'(x)+ c_0 f(x)$ has at least one zero in $[a,b]$ , where $f^{(n)}$ denotes the $n$ -th derivative of $f.$