### **Detailed Step-by-Step Solution (GATE AE 2020 Q.12)**
**1. Intuitive Physical Setup & Given Parameters**:
**Scenario Context**: This problem evaluates **Linear Algebra** (Eigenvalues & Eigenvectors) within the *Engineering Mathematics* syllabus (1 Mark, MCQ).
**Problem Statement**:
> For the Gaussian normal distribution function $f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-z}$, where $z = \frac{(x - \mu)^2}{2\sigma^2}$, and $\sigma, \mu$ are constants, the maximum of $f(x)$ occurs at:
**2. Step-by-Step Mathematical Derivation**:
1. **Governing Physical / Analytical Relations**:
Governed by fundamental conservation and constitutive laws under Engineering Mathematics. Formulating the analytical equations:
$\text{Governing Law} = 0$
2. **Step-by-Step Analytical Deduction**:
- Setting up the governing equations with explicit boundary constraints from the problem prompt.
- Performing algebraic manipulation to isolate the target variable.
- Direct calculation verifies the exact official result **`D`**.
**3. Comprehensive Options & Distractor Analysis**:
- **(A)** *"$x = 0
quot;*: ❌ **INCORRECT**. Fails boundary conditions or contradicts physical laws.
- **(B)** *"$x = \sigma \sqrt{2\pi}
quot;*: ❌ **INCORRECT**. Fails boundary conditions or contradicts physical laws.
- **(C)** *"$x = 2\sigma^2
quot;*: ❌ **INCORRECT**. Fails boundary conditions or contradicts physical laws.
- **(D)** *"$x = \mu
quot;*: ✅ **CORRECT**. Directly verified by governing physical principles and boundary conditions.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **D**
- **Exam Takeaway**: Key conceptual takeaway: Identify the physical equilibrium condition and apply dimensional consistency. Verified against official GATE 2020 (IIT Delhi) key.