Question

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 ______.

Answer 1

Correct Answer: 4

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: \( m + n = 4 \).