Find the minimum value of $1000a+\frac{1}{1000b(a-b)}$ Given that $a>b>0$ , find the least value of:- $$1000a+\frac{1}{1000b(a-b)}$$
Can anyone please help me out with this one? I tried but couldn't think of anything. Tried using A.M.G.M inequality but that gives an expression containing both $a$ and $b$ in the expression so it can't be used to find the least value i think. Please! Help! Thanks a lot!

Hint. Note that
$$1000a+\frac{1}{1000b(a-b)}=1000(a-b)+1000b+\frac{1}{1000b(a-b)}$$
then apply AM-GM inequality (this time for three terms instead of two!).
