Asubmerged submarine is stationary. the engines are put on maximum power. the submarine moves forward. the engines maintain maximum power. the forces resisting the movement of the submarine increase as its speed increases. the submarine remains horizontal. explain how the forces acting on the submarine affect its acceleration up to and after it reaches its maximum possible speed.

This is EXACTLY the same scenario as the skydiver jumping
out of the airplane, except the whole thing is turned on its side.

==> The skydiver leaves the airplane.
The force of gravity on him (his weight) makes him accelerate down.
But the air resists his downward motion.
The faster he falls, the more UPWARD force the air exerts on him.
The more upward force the air exerts, the less he accelerates down.
When his falling speed is great enough, he stops accelerating, and
falls with a constant speed.  He calls that speed his 'terminal velocity'. 

==> The submarine turns on its engines, at maximum power.
The force of the engines makes the sub accelerate forward. 
But the water resists its forward motion.
The faster it moves, the more BACKWARD force the water exerts on it.
The more backward force the water exerts, the less it accelerates forward.
When the forward speed is great enough, it stops accelerating, and moves
with a constant speed.  I don't know if they use the same term in submarines,
but you might say that speed is the 'terminal velocity' in water.