[proofplan] The shock speed is chosen so that the Rankine-Hugoniot jump relation holds by direct substitution. The main point is to identify this speed as a secant slope of the strictly convex flux $f$. The [mean value theorem](/theorems/186) places that secant slope at $f'(c)$ for some intermediate state $c \in (u_R,u_L)$, and strict convexity implies that $f'$ is strictly increasing, giving the Lax inequalities. [/proofplan] [step:Verify the Rankine-Hugoniot jump relation for the proposed speed] The jump in the state across the discontinuity is $u_R - u_L$, and the jump in the flux is $f(u_R) - f(u_L)$. By the definition of $s$, \begin{align*} s(u_R - u_L) = \frac{f(u_R) - f(u_L)}{u_R - u_L}(u_R - u_L) = f(u_R) - f(u_L). \end{align*} Since $u_L > u_R$, the denominator $u_R - u_L$ is nonzero, so the formula for $s$ is well-defined. Therefore the discontinuity moving along the line $x = st$ satisfies the Rankine-Hugoniot condition. [/step] [step:Realize the shock speed as a derivative at an intermediate state] Because $f: \mathbb{R} \to \mathbb{R}$ is continuous on $[u_R,u_L]$ and differentiable on $(u_R,u_L)$, the mean value theorem applies to $f$ on this interval. Hence there exists $c \in (u_R,u_L)$ such that \begin{align*} f'(c) = \frac{f(u_L) - f(u_R)}{u_L - u_R}. \end{align*} Multiplying numerator and denominator by $-1$ gives \begin{align*} f'(c) = \frac{f(u_R) - f(u_L)}{u_R - u_L} = s. \end{align*} Thus the Rankine-Hugoniot speed is $s = f'(c)$ for some intermediate state $c$ strictly between $u_R$ and $u_L$. [guided] The purpose of this step is to turn the shock speed into something that can be compared with the characteristic speeds $f'(u_L)$ and $f'(u_R)$. Since $s$ is a secant slope of the graph of $f$ between the two states, the mean value theorem is the correct tool. The hypotheses of the mean value theorem are satisfied on the closed interval $[u_R,u_L]$: the function $f$ is continuous there because $f \in C^2(\mathbb{R})$, and $f$ is differentiable on the open interval $(u_R,u_L)$ for the same reason. Since $u_R < u_L$, the interval is nondegenerate. Therefore there exists a point $c \in (u_R,u_L)$ such that \begin{align*} f'(c) = \frac{f(u_L) - f(u_R)}{u_L - u_R}. \end{align*} The speed in the theorem is written with both numerator and denominator reversed. These two quotients are equal because multiplying numerator and denominator by $-1$ does not change the quotient: \begin{align*} \frac{f(u_L) - f(u_R)}{u_L - u_R} = \frac{f(u_R) - f(u_L)}{u_R - u_L}. \end{align*} Hence \begin{align*} s = f'(c) \end{align*} for some $c \in (u_R,u_L)$. This is the key reduction: the shock speed is not an arbitrary number, but a characteristic speed at an intermediate state. [/guided] [/step] [step:Use strict convexity to order the characteristic speeds] Since $f''(u) > 0$ for every $u \in \mathbb{R}$, the derivative $f': \mathbb{R} \to \mathbb{R}$ is strictly increasing. Because \begin{align*} u_R < c < u_L, \end{align*} strict monotonicity gives \begin{align*} f'(u_R) < f'(c) < f'(u_L). \end{align*} Substituting $f'(c) = s$ yields \begin{align*} f'(u_R) < s < f'(u_L). \end{align*} Equivalently, \begin{align*} f'(u_L) > s > f'(u_R). \end{align*} These are precisely the Lax entropy inequalities for a scalar shock connecting the left state $u_L$ to the right state $u_R$. Together with the Rankine-Hugoniot relation verified above, this proves that the decreasing jump from $u_L$ to $u_R$ is a Lax entropy shock. [/step]