Let f(x,y)=4x^2+y^2+z^2. Use lagrange multipliers to find the point on plane 2x+3y+4x=12?

Question

Let f(x,y)=4x^2+y^2+z^2. Use lagrange multipliers to find the point on plane 2x+3y+4x=12 at which f(x,y,z) has its least value.

Thanks

Answer 1

minimize f(x,y,z) = 4x^2 + y^2 + z^2

on g(x,y,z) = 2x + 3y+ 4z - 12 = 0



∇f = λ ∇g

8x = 2λ

2y = 3λ

2z = 4λ

⇒ 6x = y and 8x = z

2x + 3(6x) + 4(8x) - 12 = 0 ........ plug into g(x,y)

x = 3/13, y = 6x = 18/13, z = 8x = 24/13



Answer: f_min = 72/13 at (3/13,18/13,24/13)

Answer 2

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Answer 3

Fx = L*Gx ..... 4 = 2L, L = 2

Fy = L*Gy......1 = 3L, L = 1/3

Fz = L*Gz......1 = 4L, L = 1/4

G(x,y,z) = c