Solution
= Solution
When $U=0$, put
$$
x_*=\frac{\langle xh^{-3}\rangle}{\langle h^{-3}\rangle}.
$$
The pressure gradient from part (c) reduces to
$$
\boxed{
p_x=\frac{12\mu V}{h^3}(x_*-x).}
$$
Because $h^{-3}>0$, $x_*$ is a <weighted mean> of points in $[0,L]$ and therefore belongs to $[0,L]$; it lies strictly inside unless the weight is singularly concentrated at an endpoint. Thus $p_x(x_*)=0$, proving that the pressure gradient vanishes somewhere between the ends.