9.8: Lagrange Multipliers
- Page ID
- 144348
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- Use the method of Lagrange multipliers to find the maximum and minimum values of \(f(x, y) = 4xy\) subject to the constraint \(\dfrac{x^2}{9} + \dfrac{y^2}{16} = 1\).
- Use the method of Lagrange multipliers to find the maximum and minimum values of \(f(x, y) = x^2 + y^2 + 2x - 2y + 1\) subject to the constraint \(x^2 + y^2 = 2\).
- Use the method of Lagrange multipliers to find the maximum and minimum values of \(f(x, y) = x^2 + y^2\) subject to the constraint \(xy = 1\).
- Use the method of Lagrange multipliers to find the maximum and minimum values of \(f(x, y) = x^2 - y^2\) subject to the constraint \(x - 2y + 6 = 0\).
- Use the method of Lagrange multipliers to find the maximum and minimum values of \(f(x, y) = 4x^3 + y^2\) subject to the constraint \(2x^2 + y^2 = 1\).
- Use the method of Lagrange multipliers to find the maximum and minimum values of \(f(x, y, z) = x + y + z\) subject to the constraint \(\dfrac{1}{x} + \dfrac{1}{y} + \dfrac{1}{z} = 1\).
- Use the method of Lagrange multipliers to find the maximum and minimum values of \(f(x, y, z) = xyz\) subject to the constraint \(x^2 + 2y^2 + 3z^2 = 6\).
- Use the method of Lagrange multipliers to find the maximum and minimum values of \(f(x, y, z) = x^2 + y^2 + z^2\) subject to the constraint \(xyz = 4\).
- Find the points on the surface \(z^2 - xy = 1\) that are closest to the origin.
- Miguel needs to construct a six-sided rectangular box with a volume of \(2\) cubic meters. What should the dimensions of the box be to minimize the box's surface area?
- A length of sheet metal is to be made into a water trough by bending up two sides of length \(x\) at an angle \(\phi\), leaving a floor of length \(y\). (See the figure below.) If the metal sheet has a width of \(2\) meters, find the values of \(x\), \(y\), and \(\phi\) that maximize the area of the trapezoid-shaped cross section. [Hint: Since the width of the sheet is \(2\) meters, \(2x + y = 2\).]
