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 \( f(x) = [x] + |x - 2| \) consists of two components: 
1. The greatest integer function, \( [x] \), which has discontinuities at integer values of \( x \). 
2. The absolute value function, \( |x - 2| \), which has a critical point at \( x = 2 \). Now, consider the interval \( -2<x<3 \). 
The points where \( f(x) \) is not continuous or differentiable are determined by: 
- Discontinuities in \( [x] \), which happen at \( x = -1, 0, 1, 2 \). 
- A critical point in \( |x - 2| \) at \( x = 2 \). So, the points where \( f(x) \) is not continuous are \( x = -1, 0, 1, 2 \), which gives us \( m = 4 \) discontinuities. The points where \( f(x) \) is not differentiable are due to the change in the slope at these points. Specifically, the function is not differentiable at \( x = 2 \), so \( n = 1 \). Thus, \( m + n = 4 + 3 = 7 \). 
Final Answer: \( m + n = 7 \).