If a consumer buys k keyboards and m mice, they receive utility u(k, m) = k 1 + 2m 1 1 . Each keyboard costs 1 and each mouse costs 8.

a) (4 points) The consumer wants to get the highest utility while spending exactly 10. Formally write the constrained optimisation problem.

b) (4 points) How many keyboards and mice should the consumer buy? Use the Lagrangian method.

c) (1 points) How would you justify with math that this point you found is indeed the maximum? (No need to do so.)

d) (1 point) Suppose the consumer now spends exactly 11 on keyboards and mice. Everything else stays the same. Without solving the problem again, approximate the new optimised utility. Justify your answer.