Let $f$ be a three times differentiable function (defined on $\mathbb{R}$ and real-valued) such that $f$ has at least five distinct real zeroes. Prove that $f+6f'+12f''+8f'''$ has at least two distinct real zeroes.
I thought about this for some time, and this is the only thing I could come up with:
Since $f$ has at least $5$ distinct real zeroes, by Rolle's Theorem $f'$ must have at least $4$ distinct zeroes. With a similar argument, we can conclude that $f''$ and $f'''$ must have at least $3$ and $2$ distinct zeroes, respectively. However, I have no idea what I could use to relate the zeroes of $f$ and the zeroes of it's derivatives, other than that they must be between each other (i.e. at least $4$ zeroes of $f'$ must be between the zeroes of $f$)
Best Answer
Hint: Take $g(x) = 8e^{x/2}f(x)$ and find $g'''$. Now, $g$ has 5 real roots and thus $g'''$ has 2 real roots atleast.