To find the least value of the expression $x^2 + 4y^2 + 3z^2 - 2x - 12y - 6z + 14$, we can complete the square for each variable term to simplify the expression. Here’s the solution:

Step 1: Completing the Square

1. For the $x$-terms: $x^2 - 2x = (x - 1)^2 - 1$

2. For the $y$-terms: $4y^2 - 12y = 4(y^2 - 3y) = 4\left((y - \frac{3}{2})^2 - \frac{9}{4}\right) = 4(y - \frac{3}{2})^2 - 9$

3. For the $z$-terms: $3z^2 - 6z = 3(z^2 - 2z) = 3\left((z - 1)^2 - 1\right) = 3(z - 1)^2 - 3$

Step 2: Substitute and Simplify

Now, rewrite the expression with the completed squares: $x^2 + 4y^2 + 3z^2 - 2x - 12y - 6z + 14 = (x - 1)^2 + 4(y - \frac{3}{2})^2 + 3(z - 1)^2 - 1 - 9 - 3 + 14$

Simplify the constants: $= (x - 1)^2 + 4(y - \frac{3}{2})^2 + 3(z - 1)^2 + 1$

Step 3: Find the Least Value

Each squared term $(x - 1)^2$, $4(y - \frac{3}{2})^2$, and $3(z - 1)^2$ is non-negative and achieves its minimum value (0) when $x = 1$, $y = \frac{3}{2}$, and $z = 1$.

Thus, the least value of the expression is: $1$

Conclusion

The correct answer is: B. 1

Would you like a more detailed breakdown or have any questions about this solution?

Related Questions

1. How does completing the square help in minimizing quadratic expressions?
2. Can the same technique be applied to expressions with more complex coefficients?
3. How would you find the maximum value of a similar expression?
4. What is the geometric interpretation of the least value in this case?
5. How does this technique relate to optimization in multivariable calculus?

Tip

Always check if completing the square can simplify the expression when finding minimum or maximum values of quadratic forms.