Question Statement:
A function f is given as:
f(X) = 4X - X2
The function f is maximized when X is equal to
Official Answer Key:
2 to 2
Step-by-Step Solution & Explanation
### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2020 Q.60)**
**1. Governing Concept & Definition**:
- **Topic**: Differential Calculus & Unconstrained Optimization
- For a concave quadratic function $f(X)$, the maximum occurs at the stationary point where $f'(X) = 0$.
**2. Detailed Step-by-Step Calculation**:
- **Function**:
$f(X) = 4X - X^2$
- **First Derivative**:
$f'(X) = 4 - 2X$
- **Setting $f'(X) = 0$**:
$4 - 2X = 0 \implies 2X = 4 \implies X = 2$
- **Second Derivative Test**:
$f''(X) = -2 < 0 \quad (\text{Strict Global Maximum})$
**3. Final Answer**:
- **Calculated Value**: **2**
- **Official GATE Accepted Range**: **2 to 2**
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