Option (I) Julia buys two goods: clothing (x) and food (y). Her utility function is u = xy + y, with MU_x=y and MU_y=x+1. She is willing to spend $180 of her income (I) on these goods. The price of clothing (x) is $20 per unit and the price of food (y) is $10.

1. Mathematically, find her optimal consumption of clothing (x*). Write down and explain the objective, constraint(s), and optimality condition(s) of her choice. Show your work.
2. If the price of clothing (x) was $12, she would buy 7 units of x. Using this information together with your answer for item (a), draw and label her demand curve for clothing (x). (Your graph does not need to have accurate scale - simply indicate the relevant values in the axes of the graph.)