Calculus

### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2022 Q.32)** **1. Governing Concept & Definition**: - **Topic**: Differential Calculus & Maxima/Minima - For a differentiable function $f(x)$, critical points satisfy $f'(x) = 0$, and local maximum occurs when $f''(x) < 0$. **2. Detailed Step-by-Step Calculation**: - **Function**: $f(x) = 3x^2 - 2x^3 \quad \text{for } x > 0$ - **First Derivative**: $f'(x) = 6x - 6x^2 = 6x(1 - x)$ - **Critical Points for $x > 0$**: $6x(1 - x) = 0 \implies x = 1$ - **Second Derivative Test**: $f''(x) = 6 - 12x \implies f''(1) = 6 - 12 = -6 < 0 \quad (\text{Local Maximum})$ - **Maximum Value**: $f(1) = 3(1)^2 - 2(1)^3 = 3 - 2 = 1$ **3. Final Answer**: - **Calculated Value**: **1** - **Official GATE Accepted Range**: **1 to 1**