I have a function $f$ whose $n$'th derivative is Lipschitz-continuous with constant $1$, and satisfies $f(0)=f'(0)=\dots=f^{(n)}(0)=0$, and $f'(1)=f''(1)=\dots = f^{(n)}(1)=0$. I'm interested in finding the largest possible value that can be attained for $f(1)$.
For example, when $n=1$, the largest value $f(1)$ can attain is $1/4$ using the function $$ f(x) = \begin{cases}\frac{1}{2} x^2, & x\leq \frac{1}{2}, \\\frac{1}{4} - \frac{1}{2} (x^2 - 1)^2, & x>\frac{1}{2}. \end{cases} $$ For n=2 I think the best value is 1/32, but I don't have a proof for that.
Any pointers would be appreciated!