Continuity and Differentiability
Differentiability of Piecewise Functions
IIT-JEE 2011 Paper 2
Grade 12
Question:
Let $f : \mathbb{R} \to \mathbb{R}$ be defined by
$$f(x) = \begin{cases} -x - \frac{\pi}{2} & x \leq -\frac{\pi}{2} \\ -\cos x & -\frac{\pi}{2} < x \leq 0 \\ x - 1 & 0 < x \leq 1 \\ \ln x & x > 1 \end{cases}$$
The number of points at which $f$ is not differentiable is:
(1) $1$
(2) $2$
(3) $3$
(4) $4$
Step-by-Step Solution
Key Concept: Check each junction. At $x = -\pi/2$: both one-sided derivatives equal $-1\checkmark$. At $x = 0$: left derivative $= \sin 0 = 0$, right derivative $= 1$ — not differentiable. At $x = 1$: left $= 1$, right $= 1/1 = 1\checkmark$. Only one bad point.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)