Putnam 2015, Problem B-1

Let $f$ be a three times differentiable function (defined on $\mathbb{R}$ and real-valued) such that $f$ has at least five distinct real zeros. Prove that $f + 6f' + 12f'' + 8f'''$ has at least two distinct real zeros.

Solution:
Let \(g(x) = e^{x/2}f(x)\). Then \(g\) has at least 5 distinct real zeroes, and by repeated applications of Rolle's theorem, \(g', g'', g'''\) have at least 4, 3, 2 distinct real zeroes, respectively. But
\[
g'''(x) = \frac{1}{8}e^{x/2}(f(x) + 6f'(x) + 12f''(x) + 8f'''(x))
\]
and \(e^{x/2}\) is never zero, so we obtain the desired result.