How do you minimize and maximize #f(x,y)=x^2+y^3# constrained to #0<x+3y<2#?
See below.
Given
We are looking for local minima/maxima.
This problem can be handled using lagrange multipliers. This can be done following the steps:
1) Describe
This can be done introducing the so called slack variables
then
2) Form the Lagrangian
3) Determine the stationary points of
This is done computing the solution to
where
so the equations which define the stationary points are
Solving for
4) Qualifying the stationary points
This can be done computing
and depending on the sign, if positive it is a local minimum and if negative a local maximum.
Ex.
The evaluations must be done according to the values found for
Attached a plot showing the region with the objective function level curves, and the stationary points showing the gradient direction.
The qualification is left to the reader.
Note. Of course if the evaluation gives zero we will continue the qualification process but this is another chapter.
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Constrain even more to
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To minimize and maximize the function ( f(x, y) = x^2 + y^3 ) constrained to ( 0 < x + 3y < 2 ), follow these steps:

Solve the constraint equation for one variable in terms of the other. Here, solve for ( x ): ( 0 < x + 3y < 2 ) Subtract ( 3y ) from each part: ( 3y < x < 2  3y )

Differentiate the objective function ( f(x, y) = x^2 + y^3 ) with respect to ( x ) and ( y ), and set the resulting expressions equal to zero to find critical points.

Evaluate the critical points within the feasible region ( 3y < x < 2  3y ) to find the values of ( x ) and ( y ) that minimize and maximize ( f(x, y) ).

Evaluate the function ( f(x, y) ) at the critical points and the boundary points of the feasible region to determine the minimum and maximum values.

Compare the values obtained in step 4 to determine the minimum and maximum 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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