Max value is obviously going to be when the particle is directly below the point A. That's because at every other point in motion, T only has to support some component of the mass of the bead, but when the bead is directly below A, the tension supports the ENTIRE mass... which is obviously greater than any component of it.
So that's why tension is greatest at that point, and the angle in this scenario is precisely theta = pi/2.
To see why min. tension occurs at
v=0, its easier to first take
v2=2ag(3sinθ−1) and rearrange it for
sinθ=6agv2+2agThen sub it into
T−mgsinθ=3amv2 to get
T−6amv2+2mag=3amv2Hence
T=6a3mv2+2mag which is a quadratic in
v. Obviously it is minimised when
v=0.