Question

𝑓(𝑥) = 2𝑥^3 + 𝑐𝑥^2 + 2𝑥

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

Answer 1

Rewrite the function in the form: $f(x)=x(2x^2+cx+2)$. One root of equation $f=0$ is $x=0$. Thus, equation $2x^2+cx+2=0$ has only one root. It means that $c=4$ and $2(x+1)^2=0$. The root of the latter is $x=-1.$ Otherwise, we receive two roots or no roots. The derivative of the function $f$ is: $f'=6x^2+8x+2$. Roots of equation $f'=0$ are: $x=\frac{-8\pm4}{12}=-1,-\frac{1}{3}$. Thus, there are two roots of equation $f'=0.$