There's often confusion betweem the functions of these three kitchen appliances -- more than any others. They don't have the same functions, so let's look at what we 've really got here.
Mixer -- This is either a hand mixer, stored in a cupoboard, or a big stand version, that sits on the counter. In some really serious kitchens, has its own little cubbyhole. Mixers are designed to mix or beat things like whipping cream, meringue, or cake mixes. They incorporate lots of air into a batter for fluff purposes. Some of the big ones have dough hooks for kneading bread, but that's really about it.
Blender -- Other than making margaritas or crushed ice for parties, a blender can make mayonnaise, gazpacho, or grind peanuts for peanut butter. It cannot beat egg whites for meringues, beat cake mixes, etc. A blender is good for emergencies, to smooth out a lumpy sauce.
Food Processor -- You can chop your veg for a stir fry, zip up a pie dough, grate cheese or cabbage, and slice potatoes for home fries. A food processor is good to make a homestyle salsa, and you may even be able to sneak a commercial cake mix by one, but don't expect to whip creme in a food processor.