a) Utility maximization means that a consumer chooses in such a way that she gets as much utility as possible. She does not choose utility directly. Utility is, instead, an indirect result of consuming a certain mix of goods. Usually, there are limitations to how she can choose. Among the baskets that she can choose, she chooses the one that gives her the highest level of utility.
b) A utility function is a mathematical expression that gives a value in numbers for, say, different combinations of goods or a certain wealth. For instance:
For two goods: U(q1,q2)=q1*q2
For wealth W:U(W)=√W
Which function to use depends on the individual we are studying.
c) The criterion for utility maximization is that the marginal rate of substitution, MRS, equals the marginal rate of transformation, MRT:
MRS = MRT
This implies that, at the point of maximization, the slope of the indifference curve is equal to the slope of the budget line.