Let $(M,g)$ be an $m$-dimensional Riemannian symmetric space and $d(\cdot,\cdot)$ be the geodesic distance function. Fix any $\alpha\in M$ and $v\in T_\alpha M$ with $\|v\|=1$. 

Q1: Define
$$
g(t)=\int_M f(t,x)e^{-f(t,x)}dx,
$$
where $f(t,x)=d^2(Exp_\alpha(tv),x).$  Question: Is $g$ second-order differentiable at $t=0$?

Answer: Here $g$ is a constant function so the answer is yes. This was given by Willie Wong.

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Q2: There was a typo in the above question. To fix it, we define
$$
h(t)=\int_M f(t,x)e^{-f(0,x)}dx,
$$
where $f(t,x)=d^2(Exp_\alpha(tv),x).$ Question: Is $h$ second-order differentiable at $t=0$?