Question

Let \[ f(x)= \begin{cases} |x|, & -\infty\lt x\lt 2 \\ |2x-4|, & 2\leq x\leq 20 \end{cases} \] \(x=a\) is a point where \(f(x)\) is continuous but not differentiable and \(x=b\) is a point where \(f(x)\) is not differentiable \((a\neq b)\). Then \(a+b=\)

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

Hint:

Modulus functions are generally not differentiable where the expression inside modulus becomes zero. Always check continuity separately at piecewise junction points.

Answer 1

The Correct Option is B

Step 1: Find points where \(|x|\) is not differentiable.
For \[ f(x)=|x|,\quad -\infty\lt x\lt 2, \] the function is not differentiable at \[ x=0 \] At \(x=0\), \[ f(x)=|x| \] is continuous but left and right derivatives are unequal.
Hence, \[ a=0 \]

Step 2: Analyze the second part of the function.
For \[ f(x)=|2x-4|,\quad 2\leq x\leq 20, \] the expression inside modulus becomes zero at \[ 2x-4=0 \] \[ x=2 \] Thus, \(f(x)\) is not differentiable at \[ x=2 \] Hence, \[ b=2 \]

Step 3: Check continuity at \(x=2\).
Left-hand limit: \[ \lim_{x\to 2^-}|x|=2 \] Right-hand limit: \[ \lim_{x\to 2^+}|2x-4|=0 \] Since \[ 2\neq 0, \] the function is discontinuous at \[ x=2 \] Thus, \(x=2\) is a point where the function is not differentiable.

Step 4: Find \(a+b\).
We have \[ a=0,\quad b=2 \] Therefore, \[ a+b=2 \]

Step 5: Final conclusion.
Hence, \[ \boxed{2} \]