Space Dynamics & Orbital Mechanics

### **Detailed Step-by-Step Solution (GATE AE 2026 Q.51)** **1. Intuitive Physical Setup & Given Parameters**: We are asked to find the minimum value of the real-valued function: $f(x) = |x| + |2x + 3|$ for $x \in \mathbb{R}$, rounded off to 1 decimal place. **2. Step-by-Step Mathematical Derivation**: 1. **Critical Transition Points**: The absolute value terms change sign at their roots: - $x = 0$ - $2x + 3 = 0 \implies x = -1.5$ 2. **Piecewise Definition of $f(x)$**: - **Region 1 ($x < -1.5$)**: Both terms are negative: $f(x) = -x - (2x + 3) = -3x - 3$ Here $f'(x) = -3 < 0$, so $f(x)$ is strictly decreasing. - **Region 2 ($-1.5 \le x \le 0$)**: First term is negative, second term is positive: $f(x) = -x + (2x + 3) = x + 3$ Here $f'(x) = +1 > 0$, so $f(x)$ is strictly increasing. - **Region 3 ($x > 0$)**: Both terms are positive: $f(x) = x + (2x + 3) = 3x + 3$ Here $f'(x) = +3 > 0$, so $f(x)$ is strictly increasing. 3. **Identify the Global Minimum**: - Since $f(x)$ decreases for $x < -1.5$ and increases for $x > -1.5$, the minimum must occur at the critical corner point: $x_{\text{min}} = -1.5$ 4. **Calculate Minimum Value**: $f(-1.5) = |-1.5| + |2(-1.5) + 3| = 1.5 + |-3 + 3| = 1.5 + 0 = 1.5$ **3. Comprehensive Options & Distractor Analysis**: - **Evaluation Criteria**: Numerical Answer Type (NAT). - **Exact Analytical Result**: $f_{\text{min}} = 1.5$. - **Rounding Requirement**: One decimal place $\implies \mathbf{1.5}$. **4. Official Master Key & Exam Takeaway**: - **Official Answer Key**: **1.5** (Acceptable Range: **1.5 to 1.5**) - **Exam Takeaway**: For sums of absolute values $\sum c_i |x - x_i|$, the function is piecewise linear and convex; its global minimum always occurs at one of the corner roots $x_i$.