So I have an HP dv6-6170us with a Radeon HD 6770M 1 GB, an Intel i7-2630QM @ 2.0 GHz, 6 GB of RAM, 1366x768 native resolution. I have switchable graphics, but have it set to Fixed mode in BIOS, and have the PowerPlay crap disabled so it never switches to the integrated Intel GPU. The laptop always runs the Radeon.
I tried playing Far Cry 3 and it's unplayable on lowest setting possible, PostFX at 0, 1280x760 resolution, DX9. So basically I'm wondering if something's wrong. I've updated my drivers, and have the AMD 12.11 Beta 11 drivers from here, as well as CAP2: http://support.amd.com/us/kbarticles/Pages/AMDCatalyst1...
I've got various screenshots for you to take a look at.
Then I see a video like this and wonder why I can't play the game. Guy has Radeon HD 6770M 2 GB(mine is 1 GB), i7-2670QM, 8 GB, and runs everything on HIGH, DX11, native resolution, etc... http://www.youtube.com/watch?v=WfbA24_Cras
^^ as before but I think your laptop just isn't good enough. I know it isn't what you want to hear but Far Cry 3 is one of if not the most demanding games on the market, and a 2.0ghz cpu and 6770M probably isn't enough.