Question

F(x) = 2x³ + cx² + 2x.

Suppose f is differentiable on R and has two roots. Show that f' has at least one root.

Answer 1

We say that $f$ has roots at $a$ and $b, (a<b),$ and since $f$ is continuous and differentiable everywhere as polynomial, it is continuous on $[a, b]$ and differentiable on $(a, b).$ So by the Rolle’s theorem, there is a number $k$ in $(a, b),$ such that $f'(k)=0.$ Therefore, $k$ is a root of $f'(x)=0.$