Question

Prove that \( f(x) = |x - 2| \) is not differentiable at \( x = 2 \).

Hint:

A function with a sharp corner or cusp, like \( |x - a| \), is not differentiable at the point where the corner occurs.

Answer 1

The function \( f(x) = |x - 2| \) is defined as: \[ f(x) = \begin{cases} x - 2 & \text{for } x \geq 2,
-(x - 2) & \text{for } x < 2. \end{cases} \] To prove that the function is not differentiable at \( x = 2 \), we need to check if the derivative from the left and right at \( x = 2 \) are the same.

Step 1: Left-hand derivative. The left-hand derivative is the derivative of \( f(x) = -(x - 2) \) for \( x < 2 \). The derivative of \( f(x) \) is: \[ \frac{d}{dx}[-(x - 2)] = -1. \] So, the left-hand derivative at \( x = 2 \) is \( -1 \).

Step 2: Right-hand derivative. The right-hand derivative is the derivative of \( f(x) = x - 2 \) for \( x \geq 2 \). The derivative of \( f(x) \) is: \[ \frac{d}{dx}[x - 2] = 1. \] So, the right-hand derivative at \( x = 2 \) is \( 1 \).

Step 3: Conclusion. Since the left-hand and right-hand derivatives at \( x = 2 \) are not equal, the function \( f(x) = |x - 2| \) is not differentiable at \( x = 2 \). Conclusion: Thus, \( f(x) = |x - 2| \) is not differentiable at \( x = 2 \).