Question

Let \( [x] \) denote the greatest integer function, and let \( m \) and \( n \) respectively be the numbers of the points, where the function \( f(x) = [x] + |x - 2| \), \( -2<x<3 \), is not continuous and not differentiable. Then \( m + n \) is equal to:

• A. \( 9 \)
• B. \( 8 \)
• C. \( 7 \)
• D. \( 6 \)

Hint:

For functions involving greatest integer functions and absolute value functions, check for discontinuities at integer points and critical points where the derivative might not exist.

Answer 1

The Correct Option is C

The function \( f(x) = [x] + |x - 2| \) comprises two parts: 1. The greatest integer function, \( [x] \), which exhibits discontinuities at integer values of \( x \). 2. The absolute value function, \( |x - 2| \), with a critical point at \( x = 2 \). Within the interval \( -2 < x < 3 \), the points where \( f(x) \) lacks continuity or differentiability are identified as follows: Discontinuities in \( [x] \) occur at \( x = -1, 0, 1, 2 \). A critical point in \( |x - 2| \) is located at \( x = 2 \). Therefore, the points where \( f(x) \) is discontinuous are \( x = -1, 0, 1, 2 \), resulting in \( m = 4 \) discontinuities. The function is not differentiable at points where its slope changes. Specifically, \( f(x) \) is not differentiable at \( x = 2 \), indicating \( n = 1 \) point of non-differentiability. Consequently, \( m + n = 4 + 1 = 5 \). The calculation in the original text erroneously stated \( n = 3 \) and \( m + n = 7 \). Correcting this, we find \( m + n = 5 \).