Preheat the oven to 400 degrees F. Butter a 12-cup muffin tin with the softened butter and set aside.

In a large bowl, combine the flour, cornmeal, baking powder, sugar, salt, and cayenne and mix well. In a separate bowl, beat together the eggs, buttermilk, and oil. Add to the dry ingredients, mixing just until it forms a batter, being careful not to overmix. Fold in the corn, cheese, and jalapenos.

Divide evenly among the 12 muffin cups and bake until golden and a toothpick inserted into the center of the muffins comes out clean, 20 to 25 minutes. Remove from the oven and let sit in the muffin tin for 5 minutes before turning out onto a wire rack. Serve hot or at room temperature.