Calculus help please - Lagrange Multipliers?

Use Lagrange multipliers to find the given extremum. Assume that x and y are positive.

Minimize f(x, y) = x^2 − y^2

Constraint: x − 6y + 105 = 0

Minimum of f(x, y,)= ? at (x, y)= ( ? )

Thank you :)

1 Answer

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  • kb
    Lv 7
    7 years ago
    Favorite Answer

    We want to minimize f(x, y) = x^2 - y^2 with constraint g(x,y) = x - 6y + 105 = 0.

    By Lagrange Multipliers, ∇f = λ∇g.

    ==> <2x, -2y> = λ<1, -6>

    ==> λ = 2x = -2y/-6, by equating like entries

    ==> y = 6x.

    Substituting this into g:

    x - 6*6x + 105 = 0

    ==> x = 3.

    So, the critical point is (x, y) = (3, 18).

    Finally, f(3, 18) = 9 - 324 = -315 is the minimum, since for instance (105, 0) also satisfies g,

    but f(105, 0) > -315.

    I hope this helps!

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