Question

Which one of the following functions has a discontinuity in the second derivative at \(x = 0\), where \(x\) is a real variable?

• A. \( f(x) = |x|^3 \)
• B. \( f(x) = x|x| \)
• C. \( f(x) = \cos(|x|) \)
• D. \( f(x) = |x|^2 \)

Hint:

Check for discontinuities in higher derivatives where the absolute value function changes its sign, typically at \(x = 0\).

Answer 1

The Correct Option is B

Step 1: Understanding the question. 
We need to find which function has a discontinuity in its second derivative at \(x = 0\). Functions involving absolute values often cause discontinuities in higher-order derivatives. 
 

Step 2: Evaluate derivatives. 
For \( f(x) = x|x| \), \[ f(x) = \begin{cases} x^2, & x > 0 \\ -x^2, & x < 0 \end{cases} \] \[ f'(x) = \begin{cases} 2x, & x > 0 \\ -2x, & x < 0 \end{cases} \] At \(x = 0\), \(f'(x)\) is continuous (both sides approach 0). \[ f''(x) = \begin{cases} 2, & x > 0 \\ -2, & x < 0 \end{cases} \] So \(f''(x)\) is discontinuous at \(x = 0\). 
 

Step 3: Conclusion. 
Hence, \(f(x) = x|x|\) has a discontinuity in the second derivative at \(x = 0\).