On an LCD display there are flourescent backlights to shine through the lcd to make the light. These backlights have a very hard-working part called the inverter. The inverter is similar to the ballast of a light fixture, except it's all done with small scale components, which run very hot and don't live very long.
I've fixed many LCD computer displays and laptop screens which had failed backlight invertors, which are typically a small separate circuit board. For example, a Compaq 20" LCD computer display, the board is approx 2" x 8", heavily protected with clear plastic, and has those blackish marks you see on a part that runs very hot.
I have not repaired a LCD TV, but I would suspect that they have multiple inverters due to the area of the screen.
My guess would be that either the connection to your backlight invertor has failed, more than one invertor has failed, or the power supply feeding these parts has lost it. Note that the invertor converts typically ~12V or so to around ~30KV, so approach with caution.
Tesla, February 2008