Please solve on paper so I can follow along and learn it

1. Find the rst- and second-order partial derivatives for the following functions. (i) (10 points) f(x, y) =eylnx (ii) (10 points) y = 102 102x'2 2 _ x Given the utility function of a consumer U = (x+1) (y+l), where x and y denote the quantities consumed of good 1 and 2, respectively. The consumer faces a budget of $100. Also, the unit price of good 1 is $1, and the unit price of good 2 is $2. (5 points) Write down the budget constraint equation. (10 points) Use Lagrange multiplier method to nd the optimal quantities demanded of the two goods and the value of the Lagrange multiplier. (5 points) If the budget constraint increases from $100 to $101, What's the approximate change in the optimal value of U