the brain also needs carbs
Mine gets plenty of both in ice cream.
Like the rest of your body, your brain will preferentially metabolize glucose.
But if blood glucose is restricted, it run just fine on b-hydroxybutyrate, acetoacetate, and acetone.
Actually, it can run on ketones just fine. Many people in the Arctic for centuries have lived their whole post-weaning lives on diets with almost no carbs and been just fine.