let \( f(x)=|x-2|+|x-3|,\ x\in R \) (Set of real numbers). Then which one of the following is TRUE?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( f(x) \) is not differentiable at both \( x=2 \) and \( x=3 \).
Removing the moduli on each interval gives
\( \displaystyle f(x)=\begin{cases}5-2x, & x\lt 2\\ 1, & 2\le x\lt 3\\ 2x-5, & x\ge 3\end{cases} \)
The function is a sum of continuous functions, so it is continuous everywhere.
The slope is \( -2 \) for \( x\lt 2 \), 0 for \( 2\lt x\lt 3 \) and 2 for \( x\gt 3 \).
The slope changes at \( x=2 \) and at \( x=3 \), so the left and right derivatives differ there.
Hence \( f \) is not differentiable at both \( x=2 \) and \( x=3 \).