Question

Analyze the function \( f(x) = |x-1| + |x-2| + |x-3| \). Determine the points where the derivative \( f'(x) \) is undefined.

• A. \( x = 1, 3 \)
• B. \( x = 1, 2, 3 \)
• C. \( x = 1.5, 2.5 \)
• D. \( x = 0 \)

Hint:

Absolute value functions are differentiable everywhere except at the point where the expression inside the modulus becomes zero.

Answer 1

The Correct Option is B

Concept: The function \(|x-a|\) is non-differentiable at \(x=a\) because the left-hand derivative and right-hand derivative are different at that point.

Step 1: Identify each absolute value term.
The given function is: \[ f(x)=|x-1|+|x-2|+|x-3| \]

Step 2: Locate possible non-differentiable points.
Each term becomes non-differentiable where its argument becomes zero: \[ x-1=0 \Rightarrow x=1 \] \[ x-2=0 \Rightarrow x=2 \] \[ x-3=0 \Rightarrow x=3 \]

Step 3: Conclude the result.
Thus, the derivative \(f'(x)\) is undefined at: \[ x=1,2,3 \]