Suppose $f : \mathbb{R} \to \mathbb{R}$ is a 3-times differentiable function and has at least 5 distinct zeros. Which of the following necessarily has 2 distinct zeros?
A. $ f + 6f' + 12f'' + 8f''' $
B. $ f''' - 3f'' + 3f' - f$
C. $f''' + 3 \sqrt 2 f'' + 6f' +2 \sqrt2 f $
My idea:
Since the function has 5 roots, according to Rolle's theorem, $f'''$ must have at least 2 roots. But how do we get to a function which has at least two distinct roots? I know though that option A can be expressed as $ g'''(x)/(e^{x/2}) $ where $g(x) = 8f(x)e^{x/2} $.