 Solution

When $U=0$, putThe pressure gradient from part (c) reduces toBecause $h^{-3}>0$, $x_{\ast }$ 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 $p_x(x_{\ast })=0$, proving that the pressure gradient vanishes somewhere between the ends.