A baseball is thrown by the center fielder (from shoulder level) to home plate where it is caught (on the fly at shoulder level) by the catcher. It traces a parabolic path. At what point does the magnitude of the vertical component of velocity have its minimum value?

At the point where the ball reaches the maximum height, right in the middle of the line that separates the two players.

Explanation:

The ball starts at shoulder level and is caught a shoulder level, so we can conclude that the thrower throw the ball upwards (this is why the ball described a parabolic path.)

The point where the vertical velocity of the ball has a minimum value is when the ball is at the maximum height.

This is because at this point the vertical velocity is equal to zero, this happens because at this point is when the ball stops going upwards and starts going down. So the velocity (that is a continuum characteristic) goes from positive values to negative values, so it must be equal to zero at a given instant, and that instant is when the velocity is minimum.

Knowing that the ball started at shoulder level and was caught a shoulder level also tells us that the maximum height point occurs right in the middle of bot players (because the parabolas are even functions).