Let f : (-2, 2) → IR be defined by \(f(x) = \begin{cases} x[x] & \quad -2<x<0\\ (x-1)[x], & \quad 0\leq x<2 \end{cases}\)
Where [x] denotes the greatest integer function. If m and n respectively are the number of points in (-2, 2) at which y = |f(x)| is not continuous and not differentiable, then m + n is equal to ______.
Where [x] denotes the greatest integer function. If m and n respectively are the number of points in (-2, 2) at which y = |f(x)| is not continuous and not differentiable, then m + n is equal to ______.
Correct Answer: 4
Solution and Explanation
Given: The piecewise function:
\( f(x) = \begin{cases} -2x, & -2 < x < -1, \\ -x, & -1 \leq x < 0, \\ 0, & 0 \leq x < 1, \\ x - 1, & 1 \leq x < 2. \end{cases} \)
- Step 1: Check for discontinuity:
Clearly, \( f(x) \) is discontinuous at \( x = -1 \). It is also non-differentiable at this point.
Thus, \( m = 1 \).
- Step 2: Check for differentiability:
Differentiate \( f(x) \):
\( f'(x) = \begin{cases} -2, & -2 < x < -1, \\ -1, & -1 < x < 0, \\ 0, & 0 < x < 1, \\ 1, & 1 < x < 2. \end{cases} \)
\( f(x) \) is non-differentiable at \( x = -1, 0, 1 \).
- Step 3: Check \( |f(x)| \):
The absolute value \( |f(x)| \) remains the same.
Thus, \( n = 3 \).
- Step 4: Calculate \( m + n \):
\( m + n = 1 + 3 = 4 \).
Final Answer: \( m + n = 4 \).
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