For the television, radio, and newspaper sales function \[N(x, y, z)=2\text{,}300x+1\text{,}400y+900z-3x^2-y^2-z^2\] find the number of sales, \(N\), if \(x=80\), \(y=16\), and \(z=20\), where \(x\), \(y\), and \(z\) are in thousands of dollars.

The manufacturing process of a company is described by the Cobb-Douglas production function \[f(x, y)=Cx^ay^b\]

1. Approximate, to two decimal places, the number of units produced when \(85\) units of labor and \(20\) units of capital are used.

2. Find the change in the level of production if the number of units is decreased by \(1\) from \(85\) units to \(84\) units, and the number of units of capital is increased by \(5\) from \(20\) to \(25\).

3. Notice that the exponents on the variables \(x\) and \(y\) add to one. Show that this function exhibits constant returns to scale.

Find the critical points of the function \[f(x, y)=\frac{3}{2}x^2+y^2+6x-8y+9\]

Find the critical points of the function \[f(x,y)=\frac{x^2}{2}+\frac{y^2}{2}-3x+9y+5xy+6\]

Find the critical points of the function \[f(x,y)=\frac{3x^2}{2}+\frac{y^4}{2}-y^2-3\]

Find the critical points of the function \[f(x,y)=4x-9y+2\]

Locate the relative extrema, if any exist, of the function \[f(x,y)=2x^2+3y^2+8x-12y+3\]

Locate the relative extrema, if any exist, of the function \[f(x, y)=\frac{2x^3}{3}+\frac{4y^3}{3}-8y^2-50x+1\]

A manufacturer markets a product in two states, \(A\) and \(B\), and, because of the different economies of the states, must price the product differently in each. (Such pricing is called price discrimination or differential pricing.) The manufacturer wishes to sell \(x\) units of the product in state \(A\) and \(y\) units of the product in state \(B\). To do so, the manufacturer must set the price in state \(A\) at \(86-\dfrac{x}{18}\) dollars and in state \(B\) at \(122-\dfrac{y}{28}\) dollars. The cost of producing all \(x+y\) items is \(45\text{,}000+4(x+y)\) dollars. How many items should be produced for states \(A\) and \(B\), respectively, to maximize the manufacturer’s profit, and what is the maximum profit?

Use the substitution method to find and classify the relative extreme points of the function \[f(x, y)=x^2-3y^2+2x+4y\] subject to the condition that \(x-2y=4\).

Use the method of Lagrange multipliers to find the minimum of the function \[f(x, y)=x^2+10y^2\] subject to the constraint \(x-y=18\).

The production of a manufacturer is given by the Cobb-Douglas production function \[f(x,y)=30x^\frac{1}{5}y^\frac{4}{5}\] where \(x\) represents the number of units of labor (in hours) and \(y\) represents the number of units of capital (in dollars) invested. Labor costs \(\$15\) per hour and there are \(8\) hours in a working day, and \(250\) working days in a year. The manufacturer has allocated \(\$4\text{,}500\text{,}000\) this year for labor and capital. How should the money be allocated to labor and capital to maximize productivity this year?

The productivity of a company (the number of units a company is able to produce) is described by the Cobb-Douglas production function \[f(x,y)=42x^\frac{3}{7}y^\frac{4}{7}\] where \(x\) represents the number of units of labor and \(y\) the number of units of capital utilized. Approximate the change in output if the number of units of labor is decreased from \(2\text{,}187\) to \(2\text{,}183\), and the number of units of capital is increased from \(128\) to \(130\).

If \(f_x(x,y)=12xy^3\), find \(f(x,y)\) by evaluating \(\displaystyle\int 12xy^3 \; dx\).

Find the volume of the space under the surface \[f(x,y)=x+4y\] and over the rectangular region bounded by \(2\leq x \leq 5\) and \(3\leq y \leq 10\). The rectangular region is pictured in Figure 6. The view is from above looking down onto the \(xy\)-plane from the \(z\)-axis.

Find the volume of the space under the surface \[f(x,y)=3x^2+6xy^2\] and over the region bounded by the curve \(g_1(x)=x^3\), the line \(g_2(x)=4x\), and between \(x=0\) and \(x=2\). The region is pictured in Figure 7. The view is from above looking down onto the \(xy\)-plane from the \(z\)-axis.

In Example 3 we used the double integral \[\int_a^b\int_{x^3}^{4x} \left(3x^2+6xy^2\right)dy\,dx\] to determine the volume of the space below the surface \[f(x,y)=3x^2+6xy^2\] Show that the same volume can be found by changing the order of integration to \(dx\, dy\).

The function \[S(x)=20\text{,}380e^{-0.25x}\] represents the number of people per week surging to shopping malls to buy a new product \(x\) weeks after it is made available to the market.

The weekly revenue realized by a company is approximated by the revenue function \[R(x, y)=350x+600y-4x^2-3y^2\] for the sale of \(x\) units per week of Product \(A\) and \(y\) units per week of product \(B\). Over the year the company produces between \(20\) and \(30\) units of product \(A\) each week and between \(80\) and \(110\) units of product \(B\) each week. Approximate this company’s average weekly revenue from products \(A\) and \(B\) over the year.