Putnam 1941 A4

Let the roots $a,b,c$ of $$f(x)=x^3 +p x^2 + qx+r$$ be real, and let $a\leq b\leq c$ . Prove that $f'(x)$ has a root in the interval $\left[\frac{b+c}{2}, \frac{b+2c}{3}\right]$ . What will be the form of $f(x)$ if the root in question falls at either end of the interval?