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:
Show Hint
- \( 9 \)
- \( 8 \)
- \( 7 \)
- \( 6 \)
The Correct Option is C
Solution and Explanation
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: \( m + n = 7 \).
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