Question

Match the following:
In the following, \( [x] \) denotes the greatest integer less than or equal to \( x \). 

 Choose the correct answer from the options given below:

• A. a - iv, b - iii, c - i, d - ii
• B. a - iii, b - ii, c - iv, d - i
• C. a - iii, b - ii, c - i, d - iii
• D. a - ii, b - iv, c - i, d - iii

Hint:

When working with absolute value functions or piecewise functions, check for continuity and differentiability at the points where the function definition changes, like at \( x = 0 \) for functions involving \( |x| \).

Answer 1

The Correct Option is C

Let's analyze each expression and match it with the correct option: 
- (a) \( x|x| \): The function \( x|x| \) is continuous everywhere, but not differentiable at \( x = 0 \) because the derivative has a discontinuity at \( x = 0 \). Thus, it is not differentiable at \( x = 0 \), corresponding to (iv). 

- (b) \( \sqrt{|x|} \): The function \( \sqrt{|x|} \) is continuous but not differentiable at \( x = 0 \). The absolute value inside the square root introduces a cusp at \( x = 0 \), so it is not differentiable at that point. Thus, the correct matching is (ii), meaning the function is differentiable in \( (-1, 1) \), except at \( x = 0 \). 

- (c) \( x + |x| \): The function \( x + |x| \) is continuous and differentiable everywhere because it is linear in both the negative and positive domains of \( x \). Thus, it is strictly increasing in \( (-1, 1) \), corresponding to (iii). 

- (d) \( |x - 1| + |x + 1| \): The function \( |x - 1| + |x + 1| \) is continuous but not differentiable at \( x = 0 \) because the absolute value functions cause a corner at that point. Thus, it is not differentiable at at least one point in \( (-1, 1) \), corresponding to (iv). Thus, the correct matching is: \[ \text{a - iii, b - ii, c - i, d - iii} \] Therefore, the correct answer is option 3.