Assume that $u$ is an entropy solution to $u_t+\partial_x(A(u))=0$, where $A$ is a non-decreasing function on $\mathbb{R}$ such that $A(0)=0$ and $$A'(u_\ell)>\gamma'(t)=\frac{A(u_\ell)-A(u_r)}{u_\ell-u_r}>A'(u_r)$$ (namely the Lax' entropy condition, where $u_\ell, u_r$ are of the usual meanings). Prove that the shocks travel from left to right if $A$ is convex.
I got frustrated after spending a long time on it (not sure if one is supposed to consider the shock speed $\gamma'(t)$, which is always non-negative since $A$ is non-decreasing). I believe that I missed something here and that there is some very simple way to solve this problem. Can someone help me? Thanks.