For each of the utility functions below, identify the conditions under which positive quantities of both goods will be consumed at the optimum (so that the nonnegativity constraints can be ignored when such conditions are met). For each of the functional forms, also identify the conditions under which the budget constraint will be binding.

u(x, y) = x^a + y^a , where a (0, 1).

u(x, y) = x^a + y^a , where a [1,).

u(x, y) = (x + y)(1 x y).