Why my ipad mini can't use back camera?

What happens when you try to use it ?

You could try closing the Camera app (I assume that it's the Camera app that you are using) via the taskbar and see if it works when you re-open it : double-click the home button to open the taskbar and then swipe or drag the Camera app's 'screen' up and off the top of the screen to close it, and click the home button to close the taskbar.

If that doesn't fix it then try a soft-reset : press and hold both the sleep and home buttons for about 10 to 15 seconds (ignore the red slider), after which the Apple logo should appear - you won't lose any content, it's the iPad equivalent of a reboot.

When did this start?

And did you try Old Toad's King Penguin's suggestions yet?

DId you try to click the camera icon on the upper right hand corner? It looks like tha back camera is covered. Do you have a casing then? Try to remove it and keep your hand away from the back camera finder. If everything fails bring the camera to apple for repair. If under warranty, they might just replace it free

You've tried both the app closing and the soft-reset, have you also tried restoring to your last backup (after first copying any purchases over to your computer's iTunes library via File > Devices > Transfer Purchases and preferably taking your own copy of any important documents and photos) ? If that doesn't fix it either then it may be a hardware issue - have you got any other camera apps on your iPad and do they work ok ?

AFter doing the backup did you do the reset device? That is in setting>general>reset>all content and setting. If all things fail then take it to the apple retail store or service provider for repair