Subtopic: Conservation Equations of Mass, Momentum & Energy
Question Statement:
The minimum value of the function f(x) = |x| + |2x + 3| for real x is ________
(rounded off to 1 decimal place).
Official Answer Key:
1.5
Step-by-Step Solution & Explanation
### **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 $x_i$.