How do you maximize and minimize #f(x,y)=e^x+e^(3y)xy# subject to #x+2y<7#?
Local minimum at
We will using the so called slack variables to transform inequality into equality relations. So considering
Find local extrema for
is transformed into an equivalent one
Forming the lagrangian
The local extrema are included in the lagrangian stationary points. This is true mainly because
The stationary points are determined finding the solutions to
The resulting equations are
This system of equations must be solved numerically using an iterative procedure like NewtonRaphson.
https://tutor.hix.ai
Calling
where
After some tries we find
The qualification is made over
Calculating
Attached a figure with the contour map and the soution point
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To maximize and minimize ( f(x, y) = e^{x} + e^{3y}  xy ) subject to ( x + 2y < 7 ), use the method of Lagrange multipliers.

Form the Lagrangian function: [ L(x, y, \lambda) = e^{x} + e^{3y}  xy + \lambda(x + 2y  7) ]

Find the partial derivatives of ( L ) with respect to ( x ), ( y ), and ( \lambda ), and set them equal to zero: [ \frac{\partial L}{\partial x} = e^{x}  y + \lambda = 0 ] [ \frac{\partial L}{\partial y} = 3e^{3y}  x + 2\lambda = 0 ] [ \frac{\partial L}{\partial \lambda} = x + 2y  7 = 0 ]

Solve the system of equations for ( x ), ( y ), and ( \lambda ).

Evaluate the critical points.

Check for maximum and minimum values by using the second partial derivative test or evaluating ( f(x, y) ) at the critical points and endpoints of the feasible region.
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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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