The following problem was presented by a guest who gave a talk at our problem-solving seminar:
Suppose $f$ is at least $3$ times differentible on $(-1,1)$, and $f,f^{'}, f^{''}, f^{'''}$ are continuous on $[-1,1]$. Moreover $|f^{'''}(x)| \leq 1$ on $[-1,1]$ and $f(-1) = f(0) = f(1) = 0$. What is the maximum value $f$ can take on $[-1,1]?$
I have been thinking about this, and my guess is that I will need to use the mean value theorem(to get bounds for $f^{'},f^{''}$) and the Landau inequality, but I don't have a good way maximize the ratio $|\frac {f^{'2}} {f^{''}}|$ given the roots condition.