Let the projectile be fired with a velocity $v$. Then, the projectile will have two components of velocities. One is horizontal component $(v\cos\theta)$ and other is vertical component $(v\sin\theta)$. As the body ascends, vertical component reduces and becomes zero at the highest point. At this point, only horizontal component $(v\cos\theta)$ remains. Then, negative vertical component of velocity gets added to the overall magnitude. So, the overall magnitude again increases and becomes $v$ during landing.
Therefore, the minimum speed will be $v\cos\theta$ at the highest point. And, the maximum speed will be $v$ during firing and landing of the projectile.
[Read: Projectile Motion]