Mashed Potatoes with corn

2 lbs thin-skinned potatoes , like Yukon Golds, scrubbed but not peeled, sliced 1/4-inch thick 3 large garlic cloves 1 1/2 cups corn kernels (fresh or frozen)[...]

3 ears fresh corn or 1 cup frozen corn 1 lb yellow potato , peeled and cut in 1 inch cubes 1 lb red potatoes, peeled and cut in 1 inch cubes 3 medium carrots[...]