91Ó°ÊÓ

Solve the following exercise by the method of Lagrange multipliers. Maximize \(x^{2}-y^{2}\), subject to the constraint \(2 x+y-3=0\)

Calculate the following iterated integrals. $$ \int_{1}^{4}\left(\int_{x}^{x^{2}} x y d y\right) d x $$

Solve the following exercise by the method of Lagrange multipliers. Find the values of \(x, y\) that minimize $$ x^{2}+x y+y^{2}-2 x-5 y $$ subject to the constraint \(1-x+y=0\).

The production function for a firm is \(f(x, y)=64 x^{3 / 4} y^{1 / 4}\), where \(x\) and \(y\) are the number of units of labor and capital utilized. Suppose that labor costs $$\$ 96$$ per unit and capital costs $$\$ 162$$ per unit and that the firm decides to produce 3456 units of goods. (a) Determine the amounts of labor and capital that should be utilized in order to minimize the cost. That is, find the values of \(x, y\) that minimize \(96 x+162 y\), subject to the constraint \(3456-64 x^{3 / 4} y^{1 / 4}=0\). (b) Find the value of \(\lambda\) at the optimal level of production. (c) Show that, at the optimal level of production, we have \(\frac{[\text { marginal productivity of labor }]}{[\text { marginal productivity of capital] }}\) $$ =\frac{[\text { unit price of labor }]}{[\text { unit price of capital }]} $$

Both first partial derivatives of the function \(f(x, y)\) are zero at the given points. Use the second-derivative test to determine the nature of \(f(x, y)\) at each of these points. If the second-derivative test is inconclusive, so state. $$ f(x, y)=\frac{1}{x}+\frac{1}{y}+x y ;(1,1) $$

Find all points \((x, y)\) where \(f(x, y)\) has a possible relative maximum or minimum. Then, use the second-derivative test to determine, if possible, the nature of \(f(x, y)\) at each of these points. If the second-derivative test is inconclusive, so state. $$ f(x, y)=-2 x^{2}+2 x y-y^{2}+4 x-6 y+5 $$

Draw the level curve of the function \(f(x, y)=x-y\) containing the point \((0,0)\).

Find the values of \(x, y\), and z- that maximize \(x y z\) subject to the constraint \(36-x-6 y-3 z=0\).

Find the values of \(x, y\), and \(z\) that maximize \(x y+3 x z+3 y z\) subject to the constraint \(9-x y z=0\).

A certain production process uses labor and capital. If the quantities of these commodities are \(x\) and \(y\), respectively, the total cost is \(100 x+200 y\) dollars. Draw the level curves of height 600,800, and 1000 for this function. Explain the significance of these curves. (Economists frequently refer to these lines as budget lines or isocost lines.)