Question

If \( f(x) \) is defined as follows: 
$$ f(x) = \begin{cases} 4, & \text{if } -\infty < x < -\sqrt{5}, \\ x^2 - 1, & \text{if } -\sqrt{5} \leq x \leq \sqrt{5}, \\ 4, & \text{if } \sqrt{5} \leq x < \infty. \end{cases} $$ 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: {Identify points where differentiability fails}
A function is differentiable at a point if its left-hand derivative (LHD) and right-hand derivative (RHD) are equal. At \( x = -\sqrt{5} \), LHD = 0 and RHD = \( 2x = -2\sqrt{5} \). Since LHD \(eq\) RHD, \( f(x) \) is not differentiable at \( x = -\sqrt{5} \). Similarly, at \( x = \sqrt{5} \), LHD = \( 2x = 2\sqrt{5} \) and RHD = 0. Thus, \( f(x) \) is not differentiable at \( x = \sqrt{5} \). Consequently, \( k = 2 \).
Step 2: {Compute \( k - 2 \)}
Given \( k = 2 \), the calculation is: \[ k - 2 = 2 - 2 = 0 \]