GATE AE 2025 • Q40
NAT • 2 Marks
Aerodynamics & Fluid Mechanics
Airfoil & Wing Theory
Subtopic: Thin Airfoil Theory & Camber Effects
Question Statement:
The maximum value of the function f(x) = (x - 1)(x - 2)(x - 3) in the domain
[0, 3] occurs at x = ______(rounded off to two decimal places).
Official Answer Key:
1.42
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2025 Q.40)**
**1. Intuitive Physical Setup & Given Parameters**:
We are asked to find the location $x$ in the domain $[0, 3]$ where the cubic polynomial:
$f(x) = (x - 1)(x - 2)(x - 3)$
attains its **maximum value** (rounded off to two decimal places).
**2. Step-by-Step Mathematical Derivation**:
1. **Expand the Polynomial**:
$f(x) = (x - 1)(x^2 - 5x + 6) = x^3 - 5x^2 + 6x - x^2 + 5x - 6$
$f(x) = x^3 - 6x^2 + 11x - 6$
2. **Find Critical Points via First Derivative**:
$f'(x) = 3x^2 - 12x + 11 = 0$
Using the quadratic formula:
$x = \frac{-(-12) \pm \sqrt{(-12)^2 - 4(3)(11)}}{2(3)} = \frac{12 \pm \sqrt{144 - 132}}{6} = \frac{12 \pm \sqrt{12}}{6}$
$x = 2 \pm \frac{2\sqrt{3}}{6} = 2 \pm \frac{\sqrt{3}}{3} = 2 \pm \frac{1}{\sqrt{3}}$
- Point 1 (Local Maximum):
$x_1 = 2 - \frac{1}{\sqrt{3}} \approx 2 - 0.57735 = 1.42265 \approx 1.42$
- Point 2 (Local Minimum):
$x_2 = 2 + \frac{1}{\sqrt{3}} \approx 2 + 0.57735 = 2.57735 \approx 2.58$
3. **Verify Function Values across Domain $[0, 3]$**:
- At boundary $x = 0$: $f(0) = (0-1)(0-2)(0-3) = -6$.
- At boundary $x = 3$: $f(3) = (3-1)(3-2)(3-3) = 0$.
- At local minimum $x_2 \approx 2.58$: $f(x_2) = (1.58)(0.58)(-0.42) \approx -0.385 < 0$.
- At local maximum $x_1 \approx 1.42$:
$f(x_1) = (1.42265 - 1)(1.42265 - 2)(1.42265 - 3) = (0.42265)(-0.57735)(-1.57735) \approx +0.3849 > 0$
4. Since $f(x_1) = +0.385$ is strictly greater than boundary values ($f(0) = -6$, $f(3) = 0$), the global maximum on $[0, 3]$ occurs at:
$x = 2 - \frac{1}{\sqrt{3}} \approx 1.42$
**3. Comprehensive Options & Distractor Analysis**:
- **Evaluation Criteria**: Numerical Answer Type (NAT).
- **Exact Analytical Result**: $2 - \frac{1}{\sqrt{3}} \approx 1.4226$.
- **Rounding Requirement**: Two decimal places $\implies \mathbf{1.42}$.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **1.42** (Acceptable Range: **1.40 to 1.44**)
- **Exam Takeaway**: Solve $f'(x) = 3x^2 - 12x + 11 = 0$ to find local extremum $x = 2 - 1/\sqrt{3} \approx 1.42$, and check interval boundaries.
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