Background:
Let $x,y$ be points in $\mathbb{R}^D$. Let $C$ be functionals from $\mathbb{R}^D$ to $\mathbb{R}$, which are strictly convex and convex respectively. Moreover, assume that $F$ is differntiable everywhere and $C$ is differentiable everywhere except at $0$.
Define the minimization problem:
$$ \underset{\gamma(0)=x,\gamma(1)=y,\gamma \in C^1([0,1],\mathbb{R}^D)}{\operatorname{argmin}} L(\gamma)=\int_a^b \|\dot{\gamma}\|_2^2 + C(\dot\gamma(t))\,dt. $$
I deduced that there exists a solution by convexity of the functional $L$ on the Banach space $C^1([0,1],\mathbb{R}^D)$, and it must be unqiue, since the sum of a convex and strictly convex functional.
Question:
How can I prove that \begin{equation} \frac{\partial^n \gamma}{\partial t^n}(t) = 0; n\geq 2 . \end{equation}
Failsafe:
(If this is false what other requirements do I need to impose on $C$ to make it true, weaker than strict convexity)?