Question

Let \( f(x)=|x-\alpha|+|x-\beta| \), where \( \alpha,\beta \) are roots of \( x^2-3x+2=0 \). Then the number of points in \( [\alpha,\beta] \) at which \( f \) is not differentiable is:

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

Hint:

Sum of absolute values: \begin{itemize} \item Non-differentiable at each kink point. \item Count roots of inside expressions. \end{itemize}

Answer 1

The Correct Option is A

Concept: Absolute value functions are not differentiable where the inside becomes zero. Step 1: {\color{red}Find roots.} \[ x^2 - 3x + 2 = 0 \Rightarrow x=1,2 \] So: \[ f(x)=|x-1|+|x-2| \] Step 2: {\color{red}Check nondifferentiable points.} Absolute value is non-differentiable at: \[ x=1, \quad x=2 \] Step 3: {\color{red}Within interval \( [1,2] \).} Both endpoints lie in interval. Hence two nondifferentiable points.