At what point in its trajectory does a batted baseball have its minimum speed?

1. at the end point
2. at the beginning point
3. at the top
4. somewhere at the middle height
Answers:
It depends on what assumptions you make.
If the ball can be assumed to change direction by 180 degrees when struck by at bat then there must be a point while being struck at which it has zero speed, by definition this must be a minimum speed.

A similar argument could also be applied at the other end if the ball is caught (again there must be a zero speed point).

If we ignore the limiting points at the ends then it depends on what assumption is made about air resistance. It this is ignored (or assumed to be negligible), then the minimum speed is at the top of the trajectory. This can be seen because energy is conserved and if no energy is lost to friction (air resistance) then the ball only interchanges energy between kinetic and potential energies, potential energy is maximum at the highest point, so kinetic energy - and hence speed - must be lowest at this point.

If air resistance is added then energy is continuously lost throughout the flight so the minimum speed would be at some point after (probably only just after) the highest point.
3
At the apex - - (the middle or top)
it was batted upwards-ish?

if it's a frictionless baseball, at the top. In real life, somewhere before the top.

if it was batted horizontally and it's frictionless, the minimum speed is at the moment it's batted (the rest is acceleration towards earth). If it's not frictionless, you can't say when the minimum speed will be.

pedants will argue that the baseball has its minimum speed when it's landed in some guy's supersized coke, but I say that's cheating.
at the top
3.
Assuming no air resistance, the horizontal velocity is always the same and at the top, the vertical velocity is zero.

In real life where there is air resistance, the horizontal velocity will slow down as time goes by, and the vertical velocity is zero.
3
Nice question..

The balls trajectory can be divided into a x (distance) and y( height) component. The y velocity is lowest at the top of the trajectory. The x velocity is zero at the end point. So I'm afraid it all depends on the angle the ball is struck.
At the top - it has no vertical velocity then, only horizontal.