4.3.E: Exercises
- Page ID
- 157596
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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. \(f(x,y) = \frac{x^2}{y^4-5}\) |
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.”
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.”
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.”
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.”
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).
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).
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 \(y>0\). |
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.
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.
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.