Question

Match the functions in Column I with their properties in Column II.
\[ \begin{array}{ll} \text{Column I} & \text{Column II}
\text{A) } |x| & \text{I. Strictly increasing and continuous in } (-1,1)
\text{B) } \sqrt{|x|} & \text{II. Continuous but not differentiable in } (-1,1)
\text{C) } x + |x| & \text{III. Differentiable in } (-1,1)
\text{D) } |x - |x|| + |x + 1| & \text{IV. Differentiable in } (-1,0), (0,1) \end{array} \] The correct match is: Identify the correct option from the following:

• A. A-III, B-V, C-II, D-I
• B. A-II, B-III, C-I, D-V
• C. A-I, B-II, C-V, D-IV
• D. A-IV, B-I, C-V, D-III

Hint:

For functions involving absolute values, analyze behavior on intervals and check continuity and differentiability at points where the expression inside the absolute value is zero.

Answer 1

The Correct Option is C

Step 1: Analyze each function in Column I
A) $|x|$: In $(-1,1)$, $f(x) = -x$ for $x<0$, $f(x) = x$ for $x \geq 0$. Derivative: $f'(x) = -1$ for $x<0$, $f'(x) = 1$ for $x>0$, not differentiable at $x = 0$. But $f'(x)>0$ where defined, so strictly increasing, and continuous.
B) $\sqrt{|x|}$: $f(x) = \sqrt{-x}$ for $x<0$, $\sqrt{x}$ for $x \geq 0$. Continuous, but derivative $\frac{1}{2\sqrt{|x|}}$ is undefined at $x = 0$.
C) $x + |x|$: $f(x) = 0$ for $x \leq 0$, $f(x) = 2x$ for $x>0$. Not differentiable at $x = 0$, strictly increasing, not differentiable in $(-1,1)$.
D) $|x - |x|| + |x + 1|$: For $x \geq 0$, $|x - x| + |x + 1| = 0 + (x + 1) = x + 1$; for $-1<x<0$, $|x - (-x)| + |x + 1| = 2x + (x + 1) = 3x + 1$. Differentiable in $(-1,0)$ and $(0,1)$, but not at $x = 0$. Step 2: Match with Column II
A) Strictly increasing and continuous: I.
B) Continuous, not differentiable in $(-1,1)$: II.
C) Strictly increasing, not differentiable: V.
D) Differentiable in $(-1,0), (0,1)$: IV. Step 3: Select the correct option
A-I, B-II, C-V, D-IV matches option (3).