If this monitor is fairly new, you should not disassemble it but return it to the dealer -- even if it's been out of warranty for, say, a year. I have monitors that are still working fine after at least 5 years and in most jurisdictions consumer rights assume you are entitled to a reasonable life from a product like this.
If you insist on DIY troubleshooting, you'll have to open the case and see -- could be the back light itself has failed. If you can read a circuit board and can isolate or remove the fuse, test it with a multimeter.
Obviously do this with the monitor power cable removed from the wall socket.
Even if it is just the fuse, fuses fail for a reason.