[proofplan] Use the Rankine-Hugoniot condition to define the shock speed as the secant slope of the flux between the two states. Strict convexity makes $f'$ strictly increasing, and the [mean value theorem](/theorems/186), equivalently the integral form of the secant slope, places that speed strictly between the two endpoint characteristic speeds. Comparing the direction of the endpoint inequalities gives exactly the compressive ordering $u_->u_+$. [/proofplan] [step:Express the shock speed as a secant slope] Assume first that $u_-\ne u_+$. For a discontinuity between the two constant states, the Rankine-Hugoniot condition gives a propagation speed $\sigma$ satisfying \begin{align*} \sigma(u_- - u_+) = f(u_-) - f(u_+). \end{align*} Thus \begin{align*} \sigma = \frac{f(u_-) - f(u_+)}{u_- - u_+}. \end{align*} This is the secant slope of $f$ across the interval with endpoints $u_-$ and $u_+$. [guided] The Rankine-Hugoniot condition is the conservation-law compatibility condition across the jump. Since the two states are distinct in a genuine discontinuity, $u_- - u_+\ne 0$, so the condition determines one speed: \begin{align*} \sigma = \frac{f(u_-) - f(u_+)}{u_- - u_+}. \end{align*} The proof is therefore reduced to comparing this secant slope with the two characteristic speeds $f'(u_-)$ and $f'(u_+)$. [/guided] [/step] [step:Place the secant slope between the endpoint characteristic speeds] Since $f''>0$ on $\mathbb R$, the derivative $f'$ is strictly increasing. By the mean value theorem, there exists $\xi$ between $u_-$ and $u_+$ such that \begin{align*} \sigma = f'(\xi). \end{align*} Because $\xi$ lies strictly between the two distinct endpoints, strict monotonicity gives \begin{align*} u_->u_+ \quad \Longrightarrow \quad f'(u_-)>\sigma>f'(u_+). \end{align*} In the opposite ordering, \begin{align*} u_-<u_+ \quad \Longrightarrow \quad f'(u_-)<\sigma<f'(u_+). \end{align*} [guided] Strict convexity is used only through the monotonicity of $f'$. The mean value theorem identifies the secant slope with one intermediate characteristic speed: \begin{align*} \sigma=f'(\xi) \end{align*} for some $\xi$ strictly between $u_-$ and $u_+$. If $u_->u_+$, then the order of the three points is $u_+<\xi<u_-$. Since $f'$ is strictly increasing, this gives \begin{align*} f'(u_+) < f'(\xi) < f'(u_-), \end{align*} or equivalently \begin{align*} f'(u_-)>\sigma>f'(u_+). \end{align*} If $u_-<u_+$, the same argument reverses the endpoint order and gives \begin{align*} f'(u_-)<\sigma<f'(u_+). \end{align*} [/guided] [/step] [step:Identify the Lax entropy condition] For a scalar convex conservation law, a jump is a Lax entropy shock precisely when the shock speed is compressive: \begin{align*} f'(u_-)>\sigma>f'(u_+). \end{align*} The previous step shows that this pair of inequalities holds exactly in the case $u_->u_+$. If $u_-<u_+$, the inequalities are reversed, so the Lax entropy condition fails. Hence the discontinuity is a Lax entropy shock if and only if $u_->u_+$. [guided] The Lax condition says that characteristics enter the shock from both sides. In this scalar setting that condition is the strict inequality \begin{align*} f'(u_-)>\sigma>f'(u_+). \end{align*} The secant-slope comparison has already shown that this inequality is equivalent to the ordering $u_->u_+$. The opposite ordering $u_-<u_+$ gives \begin{align*} f'(u_-)<\sigma<f'(u_+), \end{align*} so it is not a Lax shock. This proves the stated equivalence. [/guided] [/step]