An object with weight $W$ is dragged along a horizontal plane by force acting along a rope attached to the object. If the rope makes an angle $ \theta $ with the plane, then the magnitude of the force is
$$ F = \frac{\mu W}{\mu \sin \theta + \cos \theta} $$
where $ \mu $ is a positive constant called the coefficient of friction and where $ 0 \leqslant \theta \leqslant \pi /2 $. Show that $F$ is minimized when $ \tan \theta = \mu $.