What is the minimum value of the expression x | vedclass

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(B) Let $f(x, y, z) = x^2 + 4y^2 + 3z^2 - 2x - 12y - 6z + 14$.Rearranging the terms to complete the squares:$f(x, y, z) = (x^2 - 2x + 1) + (4y^2 - 12y

Vedclass · Accepted Answer

$1$

Vedclass · Answer

(B) Let $f(x, y, z) = x^2 + 4y^2 + 3z^2 - 2x - 12y - 6z + 14$.Rearranging the terms to complete the squares:$f(x, y, z) = (x^2 - 2x + 1) + (4y^2 - 12y + 9) + (3z^2 - 6z + 3) + 1$$= (x - 1)^2 + (2y - 3)^2 + 3(z - 1)^2 + 1$.For the expression to have a minimum value,each squared term must be zero:$(x - 1)^2 = 0 \implies x = 1$$(2y - 3)^2 = 0 \implies y = \frac{3}{2}$$(z - 1)^2 = 0 \implies z = 1$Substituting these values,the minimum value is $f(1, \frac{3}{2}, 1) = 0 + 0 + 0 + 1 = 1$.

What is the minimum value of the expression $x^2 + 4y^2 + 3z^2 - 2x - 12y - 6z + 14$?

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