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Section Practice

These are practice problems for the tests. Solutions are in the next section. While we will not present these, I am happy to answer questions about them in class.

Critical Points

  1. For each of the following functions, find the critical points and determine if they are maxima, minima, or saddle points.

    1. \(f(x,y)=1-x^{2}-y^{2}\)

    2. \(g(x,y)=e^{-x} \sin y\)

    3. \(F(x,y)=2x^{2}+2xy+y^{2}-2x-2y+5\)

    4. \(g(x,y)=x^{2}+xy+y^{2}\)

    5. \(z=8x^{3}-24xy+y^{3}\)

  2. For each of the following functions, find the absolute extrema of the function on the given closed and bounded set \(R\) in \(\re ^2\text{.}\)

    1. \(f(x,y)=2x^{2}-y^{2}\text{;}\) \(R = \{(x, y): x^{2}+y^{2}\leq 1 \}\)

    2. \(g(x,y)=x^{2}+3y^{2}-4x+2y-3\text{;}\) \(R = \{(x, y): 0 \le x \le 3\text{,}\) \(-3 \le y \le 0 \}\)

  3. Find the direction at which the maximum rate of change of \(g(x,y)= \ln(xy)-3x+2y\) at \(p=(3,2)\) will occur and find the maximum rate of change.

  4. Find the direction in which the function \(f(x,y)=x^3-y^5\) increases the fastest at the point \((2, 4)\text{.}\)

  5. For each of the following four problems, find all critical points of \(f\) and classify these critical points as relative maxima, relative minima, or saddle points using the second derivative test whenever possible.

    1. \(f(x,y)=xy^2-6x^2-3y^2\)

    2. \(\dsp f(x,y)=\frac{9x}{x^{2}+y^{2}+1}\)

    3. \(\dsp g(x,y)=x^{2}+y^{3}+\frac{768}{x+y}\)

Optimization and Lagrange Multipliers

  1. Find all extrema of \(f(x,y)=1-x^{2}-y^{2}\) subject to \(x+y=1\text{,}\) \(x\geq 0\text{,}\) and \(y\geq 0\text{.}\)

  2. Find all extrema of \(f(x,y)=1-x^{2}-y^{2}\) subject to \(x+y \leq 1\text{,}\) \(x\geq 0\text{,}\) and \(y\geq 0\text{.}\)

  3. Find the absolute extrema of the function \(g(x,y)=2 \sin(x)+5 \cos(y)\) on the rectangular region \(R = \{(x, y): 0 \le x \le 2, 0 \le y \le 5 \}\text{.}\)

  4. Find the minimum value of \(z=x^{2}+y^{2}\) subject to \(x+y=24\text{.}\)

  5. Find the extreme values of \(f(x, y) = 2x^2 +y^2-y\) subject to \(x^2+y^2=4\) using Lagrange multipliers.

  6. Find three positive numbers whose sum is 123 such that their product is as large as possible.

  7. A container in \(\re^{3}\) has the shape of a cube with each edge length 1. A (triangular) plate is placed in the container so that it intersects the cube in the plane \(x+y+z=1\text{.}\) If the container is heated so that the temperature at each point is given by \(T(x,y,z)=4-2x^{2}-y^{2}-z^{2}\) in hundreds of degrees, what are the hottest and coldest points on the plate?

  8. A company has three production plants, each manufacturing the same product. If plant A produces \(x\) units at the cost of \(\dollar(x^2+2,000)\text{,}\) plant B produces \(y\) units at the cost of \(\dollar(2y^2+3,000)\text{,}\) and plant C produces \(z\) units at the cost of \(\dollar(z^2+4,000)\text{.}\) If there is an order for 11,000 units to be filled, determine how the production should be arranged among these three plants so that the total production cost can be minimized.

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