What are the first order conditions for this problem show

Question

Suppose a consumer seeks to maximize his/hers utility function U = -(1/x₁) -(1/x₂) subject to the budget constraint p_lx₁ + p₂x₂ = Y . (Believe it or not U qualifies as a utility function).

i) Show that the conditions of the globability theorem hold so any local maximum is a global maximum.

ii) What is the Lagrangion function for this problem?

iii) The maximum value function for this problem is called the consumer's indirect utility function. Use the envelope theorem to show how this function changes when there is a change in (a) p₁ ,(b)Y.

iv) What are the first order conditions for this problem?

v) Show that the second order conditions hold.

vi) Find the consumers demand functions.

vii) Find the maximum value function and use it to check your results of (iii).