The 11 digit number must be divisible by 9, so the additional digit must be a 0 or a 9.  

And, to be divisible by 5, the last digit must be 0 or 5.

Let's do the maximum first.  

-------------------------------

The additional digit is 9, so that the number can start with 99.  So, maybe the answer is 99876543210?

It is divisible by 9 and 5, so let's check mod 11.

9-9+8-7+6-5+4-3+2-1+0 = 4, so this is not divisible by 11.  We need to rearrange these digits to reduce the check sum by 4.

Can we arrange the 2 low end digits, 10, to pick up +4?  Clearly not.  -1+0 = -1.  If we rearrange them we get 0-1 = +1, which is going in the wrong direction

What about the 3 low end digits? 2-1+0 = 1.  No way to make it -3 by rearranging terms

What about the 4 low end digits? -3+2-1+0 = -2. No way to make it -6 by rearranging terms.

What about the 5 low end digits? 4-3+2-1+0 = +2.  Hooray!  We can make it -2 by swapping the 3 and 4, and the 1 and 2.

Final answer: 99876534120.

Let's do the minimum

-------------------------

We want the extra digit to be 0, so that we can get the lowest possible.  We want the number to start 001 if leading zeroes are allowed, or 100 if they are not.  Either way, the last digit must be 5, in order to make things divisible by 0.

So, is 00123467895 divisible by 11?

0-0+1-2+3-4+6-7+8-9+5 = +1.  Any rearrangement of terms changes the sum by an even number, so we need to pick up +10 or lose 12 by rearranging terms.  A bigger change than before. And the 5 is fixed.

Working with 8-9 = -1 won't be a big enough change.

Working with 6-7+8-9 = -2 won't work either.  If we made 8 and 9 positive (ie, odd terms) and 6 and 7 negative (ie, even terms), then we only go +6 (in effect, switching 6 and 9).  If we make  6 and 7 positive (ie, odd terms) and 8 and 9 negative (ie, even terms), then we only pick up -2 (in effect, switching 7 and 8). 

Similarly, working with -4+6-7+8-9 = -6 doesn't work.  Only two plus signs to work with, so the biggest change we can effect is -4-6-7+8+9 (which only picks up 6) or +4+6-7-8-9 (which only loses 8).

Working with 3-4+6-7+8-9 = -3 does work.  +3+4+6-7-8-9 = -11, which only loses 8 (not enough).  But there is still hope.  -3-4-6+7+8+9 = 11 which picks up 14 (too much).  But we can work with too much.  We need to swap two digits of opposite signs which are two apart, and that must be 6 and 8.  -3-4+6+7-8+9 = +7, which picks up 10.  The odd digits are then 348 (in that order) and the even digits are 679 (in that order), which makes the number 00126374985.  Or, if leading zeroes are not allowed, then 10026374985.