Question

\(f(x) =   \begin{cases}     \frac{sin(x-|x|)}{x-|x|}       & \quad {x \in(-2,-1) } \\     max{2x,3[|x|]},  & \quad \text{|x|<1}\\1 & \quad \text{,otherwise}  \end{cases}\) Where [t] denotes greatest integer t. If m is the number of points where f is not continuous and n is the number of points where f is not differentiable, then the ordered pair (m, n) is

• A. (3, 3)
• B. (2, 4)
• C. (2, 3)
• D. (3, 4)

Answer 1

The Correct Option is C

 To determine the ordered pair \((m, n)\), where \(m\) is the number of points where the function \(f(x)\) is not continuous and \(n\) is the number of points where \(f(x)\) is not differentiable, we will analyze each case of the function provided:

The function \(f(x)\) is defined as:

1. \(f(x) = \frac{\sin(x - |x|)}{x - |x|}\) for \(x \in (-2, -1)\)
2. \(f(x) = \max\{2x, 3[|x|]\}\) for \(|x| < 1\)
3. \(f(x) = 1\) otherwise

Let's analyze the continuity and differentiability:

1. In the first case, \(x \in (-2, -1)\), since the function involves division by \(x - |x|\) which becomes zero at \(x = 0\) (but this is outside the given interval), we check continuity only within the interval. The given range does not include endpoints, so usually no issues of continuity differentiation arise within this open interval.
2. In the second case, \(\max\{2x, 3[|x|]\}\), simply observe the discontinuity around integers and the endpoints. Here, \([|x|]\) will lead to changes at \(x = 0\) and \(x = \pm1\). These are potential points where \(f(x)\) is not continuous or differentiable.
3. For the third case, where \(x\notin (-2, -1) \cup (-1, 1)\), the function is constant and thus continuous and differentiable.

Identifying discontinuity:

• At \(x = -1\) and \(x = 1\), there is a shift from one definition to another, check these transitions:
• At \(x = 0\): Since \(f(x)\) changes from a max of two functions, there might be issues with differentiability.

Points of discontinuity are \(x = -1\) and \(x = 1\) i.e., \(m = 2\).

For differentiability, potential non-differentiable points include where we have \(2x \neq 3[|x|]\) and removal or break points, giving \(n = 3\) at points \(-1\), \(0\), and \(1\).

Hence, the ordered pair \((m, n)\) is \((2, 3)\).