Chapter 4

Find the absolute extrema of the given function on the indicated closed and bounded set \(R\). \( f(x, y)=x y-x-3 y ; \quad R\) is the triangular region with vertices \((0,0),(0,4),\) and (5,0) .

Find the absolute extrema of the given function on the indicated closed and bounded set \(R\). \(\quad f(x, y)=x^{3}-3 x y-y^{3}\) on \(R=\\{(x, y):-2 \leq x \leq 2,-2 \leq y \leq 2\\}\).

Find the absolute extrema of the given function on the indicated closed and bounded set \(R\). \( f(x, y)=\frac{-2 y}{x^{2}+y^{2}+1}\) on \(R=\left\\{(x, y): x^{2}+y^{2} \leq 4\right\\}\).

Find three positive numbers the sum of which is 27, such that the sum of their squares is as small as possible.

Find the points on the surface \(x^{2}-y z=5\) that are closest to the origin.

Find the maximum volume of a rectangular box with three faces in the coordinate planes and a vertex in the first octant on the plane \(x+y+z=1\).

The sum of the length and the girth (perimeter of a cross-section) of a package carried by a delivery service cannot exceed 108 in. Find the dimensions of the rectangular package of largest volume that can be sent.

A cardboard box without a lid is to be made with a volume of \(4 \mathrm{ft}^{3}\). Find the dimensions of the box that requires the least amount of cardboard.

Find the point the surface \(f(x, y)=x^{2}+y^{2}+10 \quad\) nearest \(\quad\) the \(\quad\) plane \(x+2 y-z=0 .\) Identify the point on the plane.

Find the point in the plane \(2 x-y+2 z=16\) that is closest to the origin.

A company manufactures two types of athletic shoes: jogging shoes and cross-trainers. The total revenue from \(x\) units of jogging shoes and \(y\) units of cross-trainers is given by \(R(x, y)=-5 x^{2}-8 y^{2}-2 x y+42 x+102 y\), where \(x\) and \(y\) are in thousands of units. Find the values of \(x\) and \(y\) to maximize the total revenue.

Find the maximum volume of a cylindrical soda can such that the sum of its height and circumference is 120 \(\mathrm{cm} .\)

Use the method of Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraints. $$ f(x, y)=x^{2} y ; x^{2}+2 y^{2}=6 $$

Use the method of Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraints. $$ f(x, y, z)=x y z, x^{2}+2 y^{2}+3 z^{2}=6 $$

Use the method of Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraints. $$ f(x, y)=x y ; 4 x^{2}+8 y^{2}=16 $$

Use the method of Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraints. $$ f(x, y)=4 x^{3}+y^{2} ; 2 x^{2}+y^{2}=1 $$

Use the method of Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraints. $$ f(x, y, z)=x^{2}+y^{2}+z^{2}, x^{4}+y^{4}+z^{4}=1 $$

Use the method of Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraints. $$ f(x, y, z)=y z+x y, x y=1, y^{2}+z^{2}=1 $$

Use the method of Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraints. $$ f(x, y)=x^{2}+y^{2},(x-1)^{2}+4 y^{2}=4 $$

Use the method of Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraints. $$ f(x, y)=4 x y, \frac{x^{2}}{9}+\frac{y^{2}}{16}=1 $$

Use the method of Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraints. $$ f(x, y, z)=x+y+z, \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1 $$

Use the method of Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraints. $$ f(x, y, z)=x+3 y-z, x^{2}+y^{2}+z^{2}=4 $$

Minimize \(f(x, y)=x^{2}+y^{2}\) on the hyperbola \(x y=1\).

Minimize \(\quad f(x, y)=x y\) on the ellipse \(b^{2} x^{2}+a^{2} y^{2}=a^{2} b^{2}\).

Maximize \(f(x, y, z)=2 x+3 y+5 z\) on the sphere \(x^{2}+y^{2}+z^{2}=19\).

$$ \begin{aligned} f(x, y) &=x^{2}-y^{2} ; x>0, y>0 ; \\ \text { Maximize } g(x, y) &=y-x^{2}=0 \end{aligned} $$

The curve \(x^{3}-y^{3}=1\) is asymptotic to the line \(y=x\). Find the point(s) on the curve \(x^{3}-y^{3}=1\) farthest from the line \(y=x\).

Maximize \(U(x, y)=8 x^{4 / 5} y^{1 / 5} ; 4 x+2 y=12\).

Minimize \(f(x, y)=x^{2}+y^{2}, x+2 y-5=0\).

Maximize \(f(x, y)=\sqrt{6-x^{2}-y^{2}}, x+y-2=0\).

Minimize \(f(x, y, z)=x^{2}+y^{2}+z^{2}, x+y+z=1\)

Minimize \(f(x, y)=x^{2}-y^{2}\) subject to the constraint \(x-2 y+6=0\).

Minimize \(\quad f(x, y, z)=x^{2}+y^{2}+z^{2} \quad\) when \(x+y+z=9\) and \(x+2 y+3 z=20\).

Use the method of Lagrange multipliers to solve the following applied problems. A rectangular box without a top (a topless box) is to be made from \(12 \mathrm{ft}^{2}\) of cardboard. Find the maximum volume of such a box.

Use the method of Lagrange multipliers to solve the following applied problems. Find the minimum and maximum distances between the ellipse \(x^{2}+x y+2 y^{2}=1\) and the origin.

Use the method of Lagrange multipliers to solve the following applied problems. Find the point on the surface \(x^{2}-2 x y+y^{2}-x+y=0\) closest to the point (1,2,-3)

Use the method of Lagrange multipliers to solve the following applied problems. Show that, of all the triangles inscribed in a circle of radius \(R\) (see diagram), the equilateral triangle has the largest perimeter.

Use the method of Lagrange multipliers to solve the following applied problems. Find the minimum distance from point (0,1) to the parabola \(x^{2}=4 y\).

Use the method of Lagrange multipliers to solve the following applied problems. Find the minimum distance from the parabola \(y=x^{2}\) to point (0,3)

Use the method of Lagrange multipliers to solve the following applied problems. Find the minimum distance from the plane \(x+y+z=1\) to point (2,1,1).

A large container in the shape of a rectangular solid must have a volume of \(480 \mathrm{~m}^{3}\). The bottom of the container costs $$\$ 5 / \mathrm{m}^{2}$$ to construct whereas the top and sides cost $$\$ 3 / \mathrm{m}^{2}$$ to construct. Use Lagrange multipliers to find the dimensions of the container of this size that has the minimum cost.

Find the point on the line \(y=2 x+3\) that is closest to point (4,2) .

Find the point on the plane \(4 x+3 y+z=2\) that is closest to the point (1,-1,1) .

A rectangular solid is contained within a tetrahedron with vertices at \((1,0,0),(0,1,0),(0,0,1),\) and the origin. The base of the box has dimensions \(x, y,\) and the height of the box is \(z\). If the sum of \(x, y,\) and \(z\) is 1.0, find the dimensions that maximizes the volume of the rectangular solid.

By investing \(x\) units of labor and \(y\) units of \(\begin{array}{llll}\text { capital, a } & \text { watch } & \text { manufacturer } & \text { can } & \text { produce }\end{array}\) \(P(x, y)=50 x^{0.4} y^{0.6} \quad\) watches. Find the maximum number of watches that can be produced on a budget of $$\$ 20,000$$ if labor costs $$\$ 100 /$$ unit and capital costs $$\$ 200 /$$ unit. Use a CAS to sketch a contour plot of the function.

Determine whether the statement is true or false. Justify your answer with a proof . The linear approximation to the function of \(f(x, y)=5 x^{2}+x \tan (y) \quad\) at \(\quad(2, \pi)\) is given by \(L(x, y)=22+21(x-2)+(y-\pi)\)

Determine whether the statement is true or false. Justify your answer with a proof . \(\quad\left(\frac{3}{4}, \frac{9}{16}\right) \quad\) is a critical point of \(g(x, y)=4 x^{3}-2 x^{2} y+y^{2}-2\)

Sketch the function in one graph and, in a second, sketch several level curves. \(f(x, y)=e^{-\left(x^{2}+2 y^{2}\right)}\).

Sketch the function in one graph and, in a second, sketch several level curves. \(f(x, y)=x+4 y^{2}\)

Evaluate the following limits, if they exist. If they do not exist, prove it. \(\lim _{(x, y) \rightarrow(1,1)} \frac{4 x y}{x-2 y^{2}}\)