An ice-cream lover has a total of $10 to spend one evening. The price of ice-cream is $p per litre. The person's preferences for buying q litres of ice-cream, leaving a nonnegative amount $(10pq) to spend on other items, are represented by the utility function:

U(q) = q + 210 pq q(0, 10/q)

(a) Find the first-order condition for a utility maximizing quantity of ice-cream. [Hint: the square root function is a power function with exponent 1 2 .]

(b) Solve the first-order condition derived in (a) in order to express the utility maximizing quantity q as a function of p.

(c) What guarantees that your quantity q is really a maximum?