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Mathematics LibreTexts

4.3.E: Exercises

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    219722
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    4.3 Exercises

    Exercise \(\PageIndex{1}-\PageIndex{6}\)

    For problems 1 through 6, find \(f_{xx}\), \(f_{yy}\), \(f_{xy}\) and \(f_{yx}\) for the function given. Confirm that \(f_{xy} = f_{yx}\).

    1. \(f(x,y) = x^2-5y^2\)
    2. \(f(x,y) = x^4+4x^3y-6x^2y^y-4xy^3+y^4\)
    3. \(f(x,y) = 5x^2y^2\)
    4. \(f(x,y) = e^{x+6y}\)
    5. \(f(x,y) = \ln (xy + 2x - 6y)\)
    6. \(f(x,y) = \frac{x^2}{y^4-5}\)
    Exercise \(\PageIndex{7}\)

    Find the critical points of \(f(x,y) = y^3 - x^3 + 15x^2 - 12y + 12\) and use the Second Derivative Test to classify them. If the test fails, say “the test fails.”

    Exercise \(\PageIndex{8}\)

    Find the critical points of \(f(x,y) = 2xy-x^2-2y^2+6x+4\) and use the Second Derivative Test to classify them. If the test fails, say “the test fails.”

    Exercise \(\PageIndex{9}\)

    Find the critical points of \(f(x,y)=y^2-4 \ln (x) + 4x\) and use the Second Derivative Test to classify them. If the test fails, say “the test fails.”

    Exercise \(\PageIndex{10}\)

    Find the critical points of \(f(x,y)= xy - 6x^2 + 3x -y+2\) and use the Second Derivative Test to classify them. If the test fails, say “the test fails.”

    Exercise \(\PageIndex{11}\)

    The origin is a critical point for the function \(f(x,y) = x^3+y^3\), and \(D = 0\) there. That is, the Second Derivative Test fails. Use what you know about shapes of functions to decide if there is a local minimum, local maximum, or saddle point for this function at (0, 0).

    Exercise \(\PageIndex{12}\)

    The origin is a critical point for the function \(f(x,y) = 15 - x^2y^2\), and \(D = 0\) there. That is, the Second Derivative Test fails. Use what you know about shapes of functions to decide if there is a local minimum, local maximum, or saddle point for this function at (0, 0).

    Exercise \(\PageIndex{13}-\PageIndex{18}\)

    For problems 13 through 18, find all local maxima, minima, and saddle points for the function.

    13. \(f(x,y) = xy -5x^2 - 5y^2 + 33y\)
    14. \(f(x,y) = 10xy-x^2-y^2+3x\)
    15. \(f(x,y) = x^3+y^3-3xy\)
    16. \(f(x,y) = 5x^2-4xy+2y^2+4x-4y+10\)
    17. \(f(x,y) = y^2e^x+x^2\)
    18. \(f(x,y) = xy+2x-\ln (x^2y)\), for \(x>0\) and \(y>0\).
    Exercise \(\PageIndex{19}\)

    The demand functions for two products are given below. \(p_1\), \(p_2\), \(q_1\), and \(q_2\) are the prices (in dollars) and quantities for products 1 and 2.

    \[q_1 = 200+3p_1+p_2 \nonumber\]

    \[q_2=150+p_1+2p_2\nonumber\]

    a. Are these two products complementary goods or substitute goods?

    b. What is the quantity demanded for each when the price for product 1 is $20 per item and the price for product 2 is $30 per item?

    c. Write a function \(R(p_1,p_2)\) that expresses the total revenue from these two products.

    d. Find the price and quantity for each product that maximizes the total revenue.

    Exercise \(\PageIndex{20}\)

    The demand functions for two products are given below. \(p_1\), \(p_2\), \(q_1\), and \(q_2\) are the prices (in dollars) and quantities for products 1 and 2.

    \[q_1 = 350+p_1+2p_2 \nonumber\]

    \[q_2=225+p_1+p_2\nonumber\]

    a. Are these two products complementary goods or substitute goods?

    b. What is the quantity demanded for each when the price for product 1 is $20 per item and the price for product 2 is $30 per item?

    c. Write a function \(R(p_1,p_2)\) that expresses the total revenue from these two products.

    d. Find the price and quantity for each product that maximizes the total revenue.

    Exercise \(\PageIndex{21}\)

    Suppose the demand functions for two products are \(q_1 = f(p_1, p_2)\) and \(q_2 = g(p_1, p_2)\), where \(p_1\), \(p_2\), \(q_1\), and \(q_2\) are the prices (in dollars) and quantities for products 1 and 2. Consider the four partial derivatives \(\frac{\partial q_1}{\partial p_1}\), \(\frac{\partial q_1}{\partial p_2}\), \(\frac{\partial q_2}{\partial p_1}\), and \(\frac{\partial q_2}{\partial p_2}\). Tell the sign of each of these partial derivatives if

    a. the products are complementary goods.

    b. the products are substitute goods.


    This page titled 4.3.E: Exercises is shared under a CC BY 3.0 license and was authored, remixed, and/or curated by Shana Calaway, Dale Hoffman, & David Lippman (The OpenTextBookStore) via source content that was edited to the style and standards of the LibreTexts platform.