A man wants to reach point B on the opposite bank of a river flowing at a speed as shown in figure. What minimum speed relative to water should the man have so that he can reach point B? 

Let v be the speed of boatman in still water.

Resultant of v and u should be along AB. Components of $v_b$ (absolute velocity of boatman) along x and y directions are,

$v_x=u-v\sin \theta$

and $v_y=v\cos \theta$

Further, $\tan {45}^0=\frac{v_y}{v_x}$

or $1=\frac{v\cos \theta }{u-v\sin \theta }$

$v=\frac{u}{\sin \theta +\cos \theta }=\frac{u}{\sqrt{2}\sin \left(\theta +{45}^0\right)}$'

v is minimum at,

$\theta +{45}^0={90}^0 or \theta ={45}^0$

and $v_{\min }=\frac{u}{\sqrt{2}}$