Not Coolmate but here's some explanations:
You are absolutely right that gravity and tension are important in this situation & that the acrobat would be speeding up as they go downwards before slowing down as the rope swings up. What you are missing here is understanding of centripetal motion.
As the acrobat swings across they trace out part of a circle. Gravity pushes the acrobat down while tension pulls the acrobat towards the center of the circle. This results in the acrobat having the greatest speed when they are at the bottom of the circle they are tracing out as they swing across (at that point they've done all the going down they're going to do and are just about to go up).
So now we know that the point we are interested in is the bottom of the arc. Since this is part of a circular motion we can use the relationship between centripetal force and speed which Coolmate has supplied.
First, we need to find the centripetal force. Remember that this is the net force the acrobat is experiencing and is directed to the centre of the circle.
The acrobat's arm can pull down on the rope with a force of 1450 N. This means that the rope can pull up on the acrobat's arm with a force of 1450 N (note: remember Newton's 3rd law). We are interested in the forces acting on the acrobat at max speed (tension & gravity) and this is how we get the centripetal force. It's important you understand centripetal force isn't unrelated to these; centripetal force is always due to other forces.
An important note with centripetal motion questions is to be aware of what direction forces are acting in. E.g. at the bottom of a circle weight is subtracted to find centripetal force whereas at the top of a circle weight is added.
Hope this helps!
Made some slight edits, please feel free to reply with working or further questions