Limits, Continuity & Differentiability
Continuity And Differentiability
nta_abhyas_2025
Grade 12
Question:
Given $f(x) = \begin{cases} x^2e^{-x} & 0 \leq x \leq 1 \\ a \sin(x+1) \cos(2x-2) + bx^3 & 1 < x \leq 2 \end{cases}$. If $f(x)$ is differentiable at $x = 1$, then the value of $|a - b|$ is
Step-by-Step Solution
Key Concept: Apply continuity and differentiability conditions simultaneously to form a system of equations for unknown constants.
From continuity at $x = 1$: $a + b = 1$. From the given condition that $f'(1)$ exists and equals $a + b$, combined with differentiability requirements, we solve the system to obtain $a = \frac{-1}{2}$ and $b = \frac{3}{2}$, giving $a = \frac{-1}{2}$.
Correct Answer: -1