Consider the function u(x1, x2) = x 2 1 + x 2 2 . Suppose we fix x2 at some positive value, so that it is a constant for our purposes. Suppose we must choose a strictly positive value of x1, so x1 > 0. We choose x1 to maximize the function.

1. What is the x1 that maximizes the function when we restrict our choice of x1 to x1 5? Hint: Take the partial derivative of u() with respect to x1
2. Take the first-order condition with respect to x1. That is, set the partial derivative of the function with respect to x1 equal to 0. Solve for x1.
3. Is the x1 you found in part b the x1 that maximizes the function (subject to the constraints)?
4. Is the second-order condition satisfied for a maximum?
5. What about when we do not place an upper bound on the choice set?