Can I combine different wheel sizes in one mtb? Imagine a mountain bike that uses a 29er fork with 29er wheel and a 27.5 wheel at the rear. Can this be realized and what are the pros and cons?
Technically it can be done. Different wheel sizes were used over the years, starting from late 1980s in some niche touring bikes, where front wheel was significantly smaller than the rear one. Mountain bikers have used a setup of 26" wheel at the front and 24" at the back, especially for downhill (even at World Cup level) at the beginning of this century, when frames tended to be suspended ridiculously high. However with the progress in frame design, this idea lost its popularity lately.

The points raised in favour of the 26"/24" setup were:


lower bottom bracket, thus lower centre of mass meaning better control on the downhill
"flatter" head tube angle, which gives better stability at high speed and again, more control on descents
smaller wheel is easier to propell
smaller wheel means shorter spokes, which adds to its stiffness,
shorter effective top tube which helps to lean more backwards on really steep sections.


Arguments of the other side were:


lower bottom bracket could mean hitting the ground with your crank when frame suspension is fully compressed (applicable to full-suspension bikes only, but only such bikes used this trick at the time)
You need relatively higher pressure in your rear tire to make up for smaller volume of it. This in turn will give worse grip of this wheel,
Smaller wheel will generally have a smaller area of contact with the ground, which can mean worse grip, but also posibility of getting stopped by obstacles which could be overcome by your front, bigger wheel
Due to flat head and seat tube angles, going uphill, especialy on steep and technical fragments can get more tricky (obviously no one would ride uphill on their 25kg downhill bike back then, this is more applicable to today's bikes)
worse aerodynamics, but it's not what you do it for.


Most of these points are equally valid for all configurations with the larger wheel at the front. Basically the goal of such change is usually to make a bike beter on the descents, not looking at its performance on flat gruond or uphills.

Now to your specific case: note that the difference between a 29" and a 27.5" wheel is not very significant - ~6% when we look at ISO rim diameters. Therefore if you aren't too sensitive to geometry changes, you may not even feel the difference.

And obviously you need a 29er frame for this to make sense. If you decided to put a 29er fork and front wheel on a 650b frame, you would end up with a bike unsteerable on flat surfaces and exhausting to your knees. There are a few bikes on the market which use various wheel sizes, such as this Trek with a 29"/26" setup, but it's still a niche.

On the side note: a small brand called Unibike once made a cross-country bike with a 26" wheel at the front and 29" at the back. This idea seemed a bit pointless for off-road use, but they tried to forces itfor some 2 years, before letting it die.