Uploaded by warlockresearch on 18.12.2012

Transcript:

Lagrange Multipliers is a very good method to minimize or maximize using a constrain, this maxima or minima we look for are calculated based

a main equation f(x,y) that resembles the variable physical, financial, we wish to maximize or minimize respect to a constrain that is an equation that has an specific value to accomplish g(x,y) =C

The method implies to build an auxiliary equation Λ that has the main and the constrain equation substracted, the constrain equation must be multiplied by λ (this is the multiplier) we must get

this term of the constrain equal to zero g(x,y)-C = 0 ,the auxiliary equation is then ready to calculate the gradients of x, y and the multiplier λ.

Once we have our auxiliary equation built then we proceed with the steps of the method , Step 1 requires to calcualate the derivatives of the auxiliary equation Λ respect to x, y and λ in order to have the

STEP1 this help us to find the point or points where the x, y and λ have a maximum or minimum together, this maximum is constraint to the equation g(x,y)-C .

STEP 2 You then must isolate λ from equation 1 and from equation 2

equation 3 is already in terms just of x and y

STEP 3 you must make the equations 1 and 2 that are equal to λ equal to each other

this will generate an single equation in terms of x and y the term lambda disapears here

STEP 4 Substitute equation 3 isolating first x or y into the equation of STEP 3

in order to get a single equation in terms of a single variable x or y

STEP 5 Obtain the numerical value of x or y by yielding equation of STEP 4

STEP 6 Substitute te value of x or y into the constrain equation to the the other varible

using the two variables get the value substitute them into f(x,y) and get the value of the Physical, or financial variable you minimize or maximize.

a main equation f(x,y) that resembles the variable physical, financial, we wish to maximize or minimize respect to a constrain that is an equation that has an specific value to accomplish g(x,y) =C

The method implies to build an auxiliary equation Λ that has the main and the constrain equation substracted, the constrain equation must be multiplied by λ (this is the multiplier) we must get

this term of the constrain equal to zero g(x,y)-C = 0 ,the auxiliary equation is then ready to calculate the gradients of x, y and the multiplier λ.

Once we have our auxiliary equation built then we proceed with the steps of the method , Step 1 requires to calcualate the derivatives of the auxiliary equation Λ respect to x, y and λ in order to have the

STEP1 this help us to find the point or points where the x, y and λ have a maximum or minimum together, this maximum is constraint to the equation g(x,y)-C .

STEP 2 You then must isolate λ from equation 1 and from equation 2

equation 3 is already in terms just of x and y

STEP 3 you must make the equations 1 and 2 that are equal to λ equal to each other

this will generate an single equation in terms of x and y the term lambda disapears here

STEP 4 Substitute equation 3 isolating first x or y into the equation of STEP 3

in order to get a single equation in terms of a single variable x or y

STEP 5 Obtain the numerical value of x or y by yielding equation of STEP 4

STEP 6 Substitute te value of x or y into the constrain equation to the the other varible

using the two variables get the value substitute them into f(x,y) and get the value of the Physical, or financial variable you minimize or maximize.