The least value of the expression x^2 + 4y^2 + 3z^2 – 2x – 12y – 6z + 14 is

Q : The least value of the expression x2 + 4y2 + 3z2 – 2x – 12y – 6z + 14 is

(A) 0

(B) 1

(C) no least value

(D) none of these

Ans: (B)

Sol. Let f(x, y, z) = x2 + 4y2 + 3z2 – 2x –12y – 6z + 14

= (x – 1)2 + (2y – 3)2 + 3(z – 1)2 + 1

For least value of f(x, y, z)

x – 1 = 0; 2y – 3 = 0 and z – 1 = 0

x = 1; y = 3/2; z = 1

Hence least value of f(x, y, z) is f( 1, 3/2, 1) = 1