16. Max-Min Problems

Homework

  1. Consider the function \(f(x,y)=x^3-xy-x+y^2\). Find all critical points. Then use the Second Derivative Test to classify each as a local minimum, local maximum or saddle or say the test fails.

  2. Find the radius, height and volume of the largest cylinder with base on the \(xy\) plane and upper circle on the cone \(z=3-\sqrt{x^2+y^2}\).
    Work in cylindrical coordinates.

    The figure shows a cone with vertex at z = 3 on the z axis and base on the circle of radius 3 in the xy plane. Inside the cone there is a cylinder with base in the xy plane and upper edge on the cone. The cone is animated so that its radius and height oscillate. When the radius is 1 and the height is 0, the volume is 0. When the radius is 0 and the height is 3, the volume is 0. Somewhere in between these two extremes is a cylinder with maximal volume.

    1. Solve by eliminating \(z\).

    2. Solve by Lagrange multipliers.

  3. Find the dimensions and volume of the largest rectangular box with base on the \(xy\) plane and upper vertices on the cone \(z=3-\sqrt{x^2+y^2}\).
    Work in rectangular coordinates.

    The figure shows a cone with vertex at z = 3 on the z axis and base on the circle of radius 3 in the xy plane. Inside the cone there is a rectangular box with base in the xy plane and upper vertices on the cone. The box is animated so that its length and width oscillate. When the length or width become 0, the volume is 0. Somewhere in between these extremes is a rectangular box with maximal volume.

    1. Solve by Lagrange multipliers.
      HINT: Rewrite the constraint to eliminate the square root.

    2. Solve by eliminating \(z\).
      HINT: What is \(xV_x-yV_y\)?

  4. Find the point that is closest to the origin on the surface \(z=\dfrac{108}{x^4y^6}\) in the first octant. Then find the distance from the point to the origin.

    The graphic shows the surface z = 1 over the quantity x to the fourth times y to the sixth in the first octant. It also shows a line from the origin to a point on the surface.

  5. A \(118\,\text{cm}\) wire is cut into \(3\) pieces of lengths \(a\), \(b\) and \(c\). The piece of length \(a\) is bent into a square. The piece of length \(b\) is bent into a rectangle whose length is twice its width. The piece of length \(c\) is bent into a rectangle whose length is \(4\) times its width. What are \(a\), \(b\) and \(c\) which minimize the total area enclosed by all \(3\) pieces?
    Solve by using Lagrange Multipliers.
    \(50\%\) off for solving by Eliminating a Variable.
    \(50\%\) Extra Credit for solving both ways. Be sure to clearly separate the calculations.

    The animation shows a wire being cut into three uneven pieces of lengths a, b, and c. The piece of length a is folded into a square with side length s. The piece of length b is folded into a rectangle with width w and length 2w. The piece of length c is folded into a rectangle with width capital W and length 4 times capital W. We are to find values of a, b, and c which minimize or maximize the total area of the three enclosed pieces.

  6. A box with no lid is to hold \(108\,\text{cm}^3\). Its bottom costs \(\text{\textdollar} 20\) per \(\text{cm}^2\). Its left and right sides cost \(\text{\textdollar} 10\) per \(\text{cm}^2\). Its front and back cost \(\text{\textdollar} 5\) per \(\text{cm}^2\). Find the dimensions of the box with minimal cost.

    The figure shows a box with no lid. The length left to right is x. The width from front to back is y. The height is z.

  7. A quartic ellipsoid is the graph of the equation: \[ \dfrac{x^4}{a^4}+\dfrac{y^4}{b^4}+\dfrac{z^4}{c^4}=1 \] where \(a\). \(b\) and \(c\) are the intercepts.
    Find the dimensions and volume of the largest rectangular solid that can fit inside the quartic ellipsoid: \[ 16x^4+81y^4+z^4=3 \]

    The animation shows a rectangular solid inside a quartic ellipsoid. Its size is oscillating.

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Supported in part by NSF Grant #1123255

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