Question

Evaluate the following statement: \[ f(x) = \sin(|x|) - |x| \text{ is not differentiable at } x = \underline{\hspace{2cm}} \]

• A. \( x = 0 \)
• B. \( x = 1 \)
• C. \( x = -1 \)
• D. \( x = 2 \)

Hint:

When dealing with absolute value functions, check where the inside of the absolute value changes sign. These points are often where the function is non-differentiable.

Answer 1

The Correct Option is A

We are asked to determine where the function \( f(x) = \sin(|x|) - |x| \) is not differentiable. 
Step 1: Understanding the Structure of the Function The function involves the absolute value of \( x \), which is a piecewise function. To understand where this function is non-differentiable, we need to check the point where \( |x| \) changes behavior — that is at \( x = 0 \). 
Step 2: Check the Differentiability at \( x = 0 \) We know that the absolute value function \( |x| \) is defined as: \[ |x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases} \] Thus, the function becomes: - For \( x \geq 0 \), \( f(x) = \sin(x) - x \). - For \( x < 0 \), \( f(x) = \sin(-x) + x \). Now, calculate the derivatives on either side of \( x = 0 \): - For \( x > 0 \), \( f'(x) = \cos(x) - 1 \). - For \( x < 0 \), \( f'(x) = -\cos(x) + 1 \). At \( x = 0 \), the left-hand and right-hand derivatives do not match, so the function is not differentiable at \( x = 0 \). Thus, the correct answer is \( x = 0 \).