Hi,
Buderschnookie has ID'd the problem and a solution for your Alexandra
palm, aka Archontophoenix alexandrae. She mentions rubbing the scale off
first which is OK but the brownish 'fuzzy' stuff is a natural waxy
protection and should be left. Check the link she posted first to ID the
scale before proceeding.
Now to the care of your palm which is not actually a mini,it can grow
over 50 feet tall in the ground. Your palm needs to be re-potted as it
appears to have lost a lot of soil and the browned tips can indicate a
root bound condition.
The palm needs to be planted at the top of the pot. Do not add soil to
the surface. Remove the plant and using a saw or perhaps a large
serrated knife, cut two inches from the bottom of the root ball. Also
slice two inches vertically all around the ball. Add and firm enough new
soil to the bottom of the pot to bring the palm to the top. Add soil
around the edges and firm. Don't fertilize until new growth appears. If
you are in an area where this palm stays out all year you can do the
re-pot now. If it comes in for the winter, wait till spring.

HTH -_- how

--
no NEWS is good

Whew - that's a lot of work for someone as inexperienced as me! I have
actually been wondering about the pot size so will re-pot soon. Could
you perhaps point me to a tutorial on how to do this? I'm worried
about making a mistake - I wouldn't want to kill it! I don't even know
what a root ball is! I've found this definition: http://www.answers.com/topic/root-ball
but wouldn't want to make a mistake. Are you saying to trim two inches
off of the depth and breadth of whatever root clump is in the soil? Or
to trim the roots down so that they extend only two inches from the
central 'ball'? As you can tell, I have no idea what I'm doing and
don't even know what the roots will look like when I remove the palm
from its pot!

Thanks for all your help,
Paul