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Thread: Lagrangian Fucntion.

  1. #1
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    Lagrangian Fucntion.

    $\displaystyle 36x - 2x^2+48y-3y^2$

    Constraint of x+y = 15.

    Can someone check to see if i am doing this correctly. I keep coming up with the wrong answer.
    $\displaystyle L = 36x - 2x^2+48y-3y^2 - \lambda (x+y-15)$

    $\displaystyle L_x = 36 - 4x - \lambda $

    $\displaystyle L_y = 48- 6y - \lambda $

    $\displaystyle L_{\lambda} = -x- y +15 $


    $\displaystyle x= \frac{\lambda-36}{-4}$

    $\displaystyle y = \frac{\lambda-48}{-6}$

    $\displaystyle \lambda = -\frac{\lambda-36}{-4}-\frac{\lambda-48}{-6}+15$

    $\displaystyle \lambda = \frac{\lambda-36}{4}+\frac{\lambda-48}{6}+15$


    $\displaystyle \lambda = \frac{3\lambda-108}{12}+\frac{2\lambda-96}{12}+15=0$

    $\displaystyle \frac{5\lambda-204}{12}=15$

    $\displaystyle 5\lambda -204 =180$

    $\displaystyle 5\lambda = 384$

    $\displaystyle \lambda=76.8$

    I have gone wrong somewhere in here but can't figure out where.
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  2. #2
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    Quote Originally Posted by el123 View Post
    $\displaystyle 36x - 2x^2+48y-3y^2$

    Constraint of x+y = 15.

    Can someone check to see if i am doing this correctly. I keep coming up with the wrong answer.
    $\displaystyle L = 36x - 2x^2+48y-3y^2 - \lambda (x+y-15)$

    $\displaystyle L_x = 36 - 4x - \lambda $

    $\displaystyle L_y = 48- 6y - \lambda $

    $\displaystyle L_{\lambda} = -x- y +15 $


    $\displaystyle x= \frac{\lambda-36}{-4}$

    $\displaystyle y = \frac{\lambda-48}{-6}$

    $\displaystyle \lambda = -\frac{\lambda-36}{-4}-\frac{\lambda-48}{-6}+15$

    $\displaystyle \lambda = \frac{\lambda-36}{4}+\frac{\lambda-48}{6}+15$


    $\displaystyle \lambda = \frac{3\lambda-108}{12}+\frac{2\lambda-96}{12}+15=0$

    $\displaystyle \frac{5\lambda-204}{12}=15$

    $\displaystyle 5\lambda -204 =180$

    $\displaystyle 5\lambda = 384$

    $\displaystyle \lambda=76.8$

    I have gone wrong somewhere in here but can't figure out where.
    I assume you're talking about Lagrange multipliers. Set f(x, y) = 36x - 2x^2 + 48y - 3y^2, g(x, y) = x + y. Taking grad(f) = lambda*grad(g), we have

    36 - 4x = lambda,
    48 - 6y = lambda,
    x + y = 15.

    Solve this system of equations.
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