Since you just bought this last weekend it may be something that was initiated before you brought it home. Perhaps it was allowed to dry out at the nursery.
Also, I have read where Cordylines are sensitive to fluoride; I really do not know if this is accurate but if your municipal water supply has high levels of it you may want to consider another source. I know rain water is scarce in your area but there are faucet water filters that remove fluoride.
Good luck and Welcome
