Camera does not respond
I see you can't even look on the screen, because you have to start the camera first, before you can switch between taking pictures and viewing. You you not even can see an error displayed on the screen.
Most obvious is the battery, or its connections. The manual says:
"The camera does not work even when batteries are installed."
Insert recharged batteries in the correct direction. "Inserting the battery and the SD/SDHC/SDXC memory card (sold separately)" (p. 13), "Charging the battery and setting up using the supplied CD" (p. 14), "Charging the battery with the supplied USB-AC adapter" (p. 17)
Battery performance may have been reduced temporarily due to low temperature. Take the batteries out of the camera, and warm them by putting them in your pocket for a while.
Then to make sure it is not the battery they advice to connect an external USB-AC adapter.:
An F-3AC USB-AC adapter (available separately) is recommended if the camera will be used for extended periods.
Note that the power cable supplied with the USB-AC adapter is for use exclusively with the USB-AC adapter. Do not use with other devices.
With the adapter, the voltage (and current it can deliver) could be enough to switch on the camera.
If it works, you should take aout the battery and check if the connection point of the battery and in the camera are clean and shiny. If not, please clean them with a dry cloth. Never use a sharp tool to clean electronic contacts. The manual says the battery will live up to 300 charge / discharge cycles. If the camera won't work with the battery and it works with the USB AC adapter, replace the battery.