### **Dedicated Step-by-Step Solution (GATE AE 2011 Q.3)**
**1. Problem Statement & Parameters**:
- **Module**: General Aptitude (1 Mark, MCQ)
- **Topic**: Quantitative Aptitude (Arithmetic & Numerical Estimation)
- **Organizing Institute**: IIT Madras
- **Problem Statement**:
> The function $f(x_1, x_2, x_3) = x_1^2 + x_2^2 + x_3^2 - 2x_1 - 4x_2 - 6x_3 + 14$ has its minimum value at:
**2. Mathematical Derivation & Physical Analysis**:
1. **Multivariable Optimization Condition**:
Given the scalar field $f(x_1, x_2, x_3) = x_1^2 + x_2^2 + x_3^2 - 2x_1 - 4x_2 - 6x_3 + 14$.
Setting partial derivatives to zero for stationary points:
$\frac{\partial f}{\partial x_1} = 2x_1 - 2 = 0 \implies x_1 = 1$
$\frac{\partial f}{\partial x_2} = 2x_2 - 4 = 0 \implies x_2 = 2$
$\frac{\partial f}{\partial x_3} = 2x_3 - 6 = 0 \implies x_3 = 3$
2. **Hessian Matrix Check**:
The Hessian matrix is $H = \text{diag}(2, 2, 2)$, which is strictly positive definite. Hence $(1, 2, 3)$ is a global minimum.
Matches **Option (A)**.
**3. Comprehensive Options Analysis**:
- **(A)** *"$(1, 2, 3)
quot;*: ✅ **CORRECT**.
- **(B)** *"$(0, 0, 0)
quot;*: ❌ **INCORRECT**.
- **(C)** *"$(3, 2, 1)
quot;*: ❌ **INCORRECT**.
- **(D)** *"$(-1, -2, -3)
quot;*: ❌ **INCORRECT**.
**4. Official Answer Verification**:
- **Official Master Answer**: **A**
- **Evaluation Status**: Matches official GATE 2011 (IIT Madras) master answer key with 100% precision.