Let $\{f_n\}$ be a sequence of (real-valued) smooth convex functions on $[0,1]$, with $f_n(0) = f_n(1) = 0$ for all $n$.

Let $t_n \in [0,1]$ be the minimizer of $f_n$ and assume that $M_n:= f_n(t_n) \to -\infty$.

Intuitively, it seems that that $V_n^2 := f''_n(t_n)$ should go to $\infty$. Can we obtain bounds on the relative growth of $V_n$ and $-M_n$? In particular, can we assure that $V_n/(-M_n) = o(1)$? (I mean, what further conditions one needs for this to hold?)

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