Problem 3
Show that the function \(y=x+1 / x\) (with \(x \neq 0\) ) has two relative extrema, one a maximum and the other a minimum. Is the "minimum" larger or smaller than the "maximum"? How is this paradoxical result possible?
Problem 4
Given the function \(y=a-\frac{b}{c+x} \quad(a, b, c>0 ; x \geq 0),\) determine the general shape of its graph by examining ( \(a\) ) its first and second derivatives, (b) its vertical intercept, and (c) the limit of \(y\) as \(x\) tends to infinity. If this function is to be used as a consumption function, how should the parameters be restricted in order to make it economically sensible?
Problem 4
Let \(T=\phi(x)\) be a total function (e.g., total product or total cost): (a) Write out the expressions for the marginal function \(M\) and the average function \(A\). (b) Show that, when \(A\) reaches a relative extremum, \(M\) and \(A\) must have the same value. (c) What general principle does this suggest for the drawing of a marginal curve and an average curve in the same diagram? (d) What can you conclude about the elasticity of the total function \(T\) at the point where \(A\) reaches an extreme value?
Problem 5
A quadratic profit function \(\pi(Q)=h Q^{2}+j Q+k\) is to be used to reflect the following assumptions: (a) If nothing is produced, the profit will be negative (because of fixed costs). (b) The profit function is strictly concave. (c) The maximum profit occurs at a positive output level \(Q^{*}\). What parameter restrictions are called for?
Problem 5
Draw the graph of a function \(f(x)\) such that \(f^{\prime}(x) \equiv 0,\) and the graph of a function \(g(x)\) such that \(g(3)=0 .\) Summarize in one sentence the essential difference between \(f(x)\) and \(g(x)\) in terms of the concept of stationary point.
