= Solution
The <Maxwell-Boltzmann velocity distribution> is unaffected by the position-dependent potential, since the canonical density factors. Its speed density is proportional to $v^2e^{-mv^2/(2k_BT)}$ for $v\ge0$. For $v>0$, differentiating its logarithm gives $2/v-mv/(k_BT)=0$, so
$$
\boxed{v_{\rm most\ probable}=\sqrt{\frac{2k_BT}{m}}.}
$$
The density vanishes at zero and at infinity and has only this positive stationary point, so it is the maximum. It is the same at every height.
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