Question

If \( f(x) \) is defined as follows:
\[ f(x) = \begin{cases} 4, & {if } -\infty < x < -\sqrt{5}, \\ x^2 - 1, & {if } -\sqrt{5} \leq x \leq \sqrt{5}, \\ 4, & {if } \sqrt{5} \leq x < \infty. \end{cases} \] If \( k \) is the number of points where \( f(x) \) is not differentiable, then \( k - 2 = \)

• A. 2
• B. 1
• C. 0
• D. 3

Hint:

For piecewise functions, check the left-hand and right-hand derivatives at the boundary points to determine non-differentiability.

Answer 1

The Correct Option is C

Step 1: {Check the points of non-differentiability}
We know that for a function to be differentiable at a point, both the left-hand derivative (LHD) and right-hand derivative (RHD) must be equal at that point. At \( x = -\sqrt{5} \), the left-hand derivative is 0, but the right-hand derivative is \( 2x = -2\sqrt{5} \). Therefore, \( f(x) \) is not differentiable at \( x = -\sqrt{5} \). Similarly, at \( x = \sqrt{5} \), the left-hand derivative is \( 2x = 2\sqrt{5} \), and the right-hand derivative is 0, meaning \( f(x) \) is not differentiable at \( x = \sqrt{5} \). Thus, \( k = 2 \). 
Step 2: {Calculate \( k - 2 \)}
Since \( k = 2 \), we find: \[ k - 2 = 2 - 2 = 0 \]