How do you maximize and minimize #f(x,y)=xyy^2# constrained to #0<x+3y<4#?
Using the Lagrange Multipliers technique the relative minima/maxima to this problem can be determined.
Those are
Attached a plot showing those points located at the feasible region frontier.
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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.

Form the Lagrangian function: [ L(x, y, \lambda) = xy  y^2 + \lambda(4  x  3y) ]

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 ]

Solve the system of equations to find critical points.

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

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

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

Form the Lagrangian function: [ L(x, y, \lambda) = xy  y^2 + \lambda(x + 3y  4) ]

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 ]

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

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

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

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.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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