Intermediate Microeconomics and Its Application 12th Edition by Walter Nicholson,Christopher Snyder

Exercise 18

Beth is a mathematical whiz. She has been reading this chapter and remarks, "All this short-run/long-run stuff is a trivial result of the mathematical fact that the minimum value for any function must be as small as or smaller than the minimum value for the same function when some additional constraints are attached." Use Beth's insight to explain the following:
a. Why short-run total costs must be equal to or greater than long-run total costs for any given output level
b. Why short-run average cost must be equal to or greater than long-run average cost for any given output level
c. That you cannot make a definite statement about the relationship between short-run and long-run marginal cost