= Solution
For real square-integrable $f,g$, nonnegativity of $\int(f-\lambda g)^2dx$ gives a quadratic in the real number $\lambda$. If $\int g^2dx>0$, choose $\lambda=\int fg\,dx/\int g^2dx$ to obtain
$$
\left(\int fg\,dx\right)^2\leq\left(\int f^2dx\right)\left(\int g^2dx\right).
$$
If $\int g^2dx=0$, then $g=0$ almost everywhere and the same inequality is immediate. This proves the integral <Cauchy-Schwarz inequality>.
Put $T=t_f-t_i>0$ and $L=x_f-x_i>0$. The forward <speed> satisfies $L=\int_{t_i}^{t_f}v\,dt$, so $V_t=L/T$. The <change of variables> $dx=v\,dt$ gives $V_x=L^{-1}\int_{t_i}^{t_f}v^2dt$. Apply the <Cauchy-Schwarz inequality> to $v$ and the constant function one:
$$
L^2\leq T\int_{t_i}^{t_f}v^2dt,
\qquad\boxed{V_x\geq V_t.}
$$
Equality holds exactly when $v$ is constant almost everywhere. Equivalently, the <time and distance averages of speed> satisfy $V_x-V_t=\operatorname{Var}_t(v)/V_t$, which identifies the excess as a measure of speed variation.
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