= Solution
The tangent-line inequality for a differentiable <concave function> is $f(r)\le f(s)+f'(s)(r-s)$. Taking $r=u=U_x$ and using the <Hamilton-Jacobi equation> yields
$$
U_t+f'(s)U_x=f'(s)u-f(u)\ge sf'(s)-f(s).
$$
Along $x(\tau)=x_0+\tau f'(s)$, the <chain rule> identifies the left side with $dU(\tau,x(\tau))/d\tau$. Integrating gives
$$
\boxed{U(t,x_0+t f'(s))-U(0,x_0)\ge t\bigl(sf'(s)-f(s)\bigr).}
$$
Strict <concavity> makes equality possible exactly when $u=s$ along the line, which will select the maximizing <characteristic curve>.
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