Lagrange multiplier help thanks?

Find the minimum value of

Px² +Qy² + Rz² + 2yz + 2xz + 2xy

Where P, Q, R are constants greater than 1

and x, y, z are constrained by: x + y + z = 1

1 Answer

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  • kb
    Lv 7
    1 decade ago
    Favorite Answer

    Let f(x,y) = Px² +Qy² + Rz² + 2yz + 2xz + 2xy subject to g(x,y,z) = x + y + z = 1.

    ∇f = λ ∇g ==> <2Px + 2z + 2y, 2Qy + 2z + 2x, 2Rz + 2y + 2x> = λ<1, 1, 1>.

    So, 2Px + 2z + 2y = 2Qy + 2z + 2x = 2Rz + 2y + 2x = λ

    ==> Px + z + y = Qy + z + x = Rz + y + x = λ/2

    ==> (P - 1)x + 1 = (Q - 1)y + 1 = (R - 1)z + 1 = λ/2, since x + y + z = 1.

    ==> (P - 1)x = (Q - 1)y = (R - 1)z = λ/2 - 1.

    Since y = (P - 1)x/(Q - 1) and z = (P - 1)x/(R - 1)

    So, x + y + z = x + (P - 1)x/(Q - 1) + (P - 1)x/(R - 1) = 1

    ==> x = 1 / [1 + (P - 1)/(Q - 1) + (P - 1)/(R - 1)].

    Now you can solve for y and z and find the minimal value.

    I hope this helps!

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