= Solution
Because $f'$ is a decreasing bijection, its inverse $h$ exists and is continuous. For any $y$, choose $s=h((x-y)/t)$ in the preceding inequality. The line from $(0,y)$ then reaches $(t,x)$, giving $U(t,x)\ge U(0,y)+t g((x-y)/t)$.
Now let $x_0=X_t^{-1}(x)$ and $s=u_0(x_0)$. On this <characteristic curve>, $u=s$, so the tangent-line inequality is an equality throughout. This proves attainment and
$$
\boxed{U(t,x)=\max_{y\in\mathbb R}\left\{U(0,y)+t g\left(\frac{x-y}{t}\right)\right\},\qquad u(t,x)=h\left(\frac{x-x_0}{t}\right).}
$$
In fact the maximizing foot is unique. The <concave Legendre dual> can be written $g(z)=\inf_s(zs-f(s))$ and satisfies $g'(z)=h(z)$: compare the minimizing values at $z$ and $z+\varepsilon$ and use continuity of $h$. This avoids assuming differentiability of $h$, which strict <concavity> by itself does not guarantee. Differentiating the maximizing expression with respect to $y$ gives $u_0(y)-h((x-y)/t)=0$, hence $X_t(y)=x$ and $y=x_0$. The <maximum representation for a concave conservation law> therefore reproduces the <characteristic solution of a scalar conservation law> throughout the smooth lifespan.
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