Find the maximum and minimum of Qx x12 2x22 3x32 in the r

Find the maximum and minimum of Q(x) = x_1^2 + 2x_2^2 + 3x_3^2 in the region D = {x R^3: x^T x = x_1^2 + x_2^2 + x_3^2 = 1}, and find a unit vector at which the maximum and the minimum are attained, respectively.

Solution

Q(x₁, x₂, x₃) = x₁² + 2x₂² +3x₃²

Q(x₁) = 2x₁

Q(x₂) = 4x₂

Q(x₃) = 6x₃

g(x₁, x₂ , x₃) = x₁² + x₂² + x₃² = 1 -------- (1)

g(x₁ ) = 2x₁

g(x₂) = 2x₂

g(x₃) = 2x₃

Using Lagrange multipliers,

Qx₁ =   gx₁

2x₁ =   2x₁

Qx₂ =   gx₂

4x₂ =   2x₂

Qx₃ =   gx₃

6x₃ =  2x₃

Lets consider the cases,

case1 :   = 0

x1 = x2 = x3 = 0

case 2 :   not equal to zero,

x1 , x2 and x3 have finite values,

a) if x1 = 0

Then by 1

x2 = +-1

x3 = +-1

critical points are : (0,1,1) or(0,1, -1)

But no such critical points exists for solution ,

Hence, we are considering (1,1,1) as critical point

Max value = 1

Min value =1

vector = i + j + k