How do you maximize and minimize #f(x,y)=xy-y^2# constrained to #0

Question

How do you maximize and minimize #f(x,y)=xy-y^2# constrained to #0<x+3y<4#?

Paisley Johnson · Accepted Answer

#((x,y,f(x,y),"type"),(0,0,0,"maximum"),(1,2.5,0.5,"maximum"))#

Using the Lagrange Multipliers technique the relative minima/maxima to this problem can be determined.

Those are

#((x,y,f(x,y),"type"),(0,0,0,"maximum"),(1,2.5,0.5,"maximum"))#

Attached a plot showing those points located at the feasible region frontier.

Christopher Agresta · Answer

undefined To maximize and minimize $ f(x, y) = xy - y^2 $ subject to the constraint $ 0 < x + 3y < 4 $, we can use the method of Lagrange multipliers.

1. Form the Lagrangian function:
$$ L(x, y, \lambda) = xy - y^2 + \lambda(4 - x - 3y) $$

2. Find the partial derivatives of $ L $ with respect to $ x $, $ y $, and $ \lambda $, and set them equal to zero:
$$ \frac{\partial L}{\partial x} = y - \lambda = 0 $$
$$ \frac{\partial L}{\partial y} = x - 2y - 3\lambda = 0 $$
$$ \frac{\partial L}{\partial \lambda} = 4 - x - 3y = 0 $$

3. Solve the system of equations to find critical points.

4. Substitute the solutions back into the original function $ f(x, y) $ to find the maximum and minimum values.

5. Check the boundary of the constraint region $ 0 < x + 3y < 4 $ for potential maximum and minimum values.

6. Evaluate the function $ f(x, y) $ at any critical points and points on the boundary to determine the maximum and minimum values.

This method will give us the maximum and minimum values of $ f(x, y) $ subject to the given constraint.

Christopher Allers · Answer

undefined To maximize and minimize $ f(x,y) = xy - y^2 $ constrained to $ 0 < x + 3y < 4 $, you can use the method of Lagrange multipliers.

1. Form the Lagrangian function:
   $$ L(x, y, \lambda) = xy - y^2 + \lambda(x + 3y - 4) $$

2. Find the partial derivatives of $ L $ with respect to $ x $, $ y $, and $ \lambda $, and set them equal to zero:
   $$ \frac{\partial L}{\partial x} = y + \lambda = 0 $$
   $$ \frac{\partial L}{\partial y} = x - 2y + 3\lambda = 0 $$
   $$ \frac{\partial L}{\partial \lambda} = x + 3y - 4 = 0 $$

3. Solve this system of equations to find critical points $ (x, y) $ and $ \lambda $.

4. Substitute these critical points into the objective function $ f(x,y) = xy - y^2 $ to find the maximum and minimum values.

5. Check the endpoints of the interval $ 0 < x + 3y < 4 $ to see if they give larger or smaller values than the critical points.

6. Compare the values obtained in step 4 and step 5 to determine the maximum and minimum values of $ f(x,y) $ subject to the given constraint.

How do you maximize and minimize #f(x,y)=xy-y^2# constrained to #0&lt;x+3y&lt;4#?

How do you maximize and minimize #f(x,y)=xy-y^2# constrained to #0<x+3y<4#?