Q. Function $f(x) = |x - 1| + |x - 2|,\ x \in R$ is
A) Differentiable everywhere in $R$
B) Except $x = 1$ and $x = 2$ differentiable everywhere in $R$
C) Not continuous at $x = 1$ and $x = 2$
D) Increasing in $R$
Explanation
(b)
Given, $f(X) = |x - 1| + |x - 2|$
$= \left\{ \begin{array}{r}
x - 1 + x - 2,\ \ \ \ x \geq 2 \\
x - 1 + 2 - x,\ \ \ 1 \leq x < 2 \\
1 - x + 2 - x,\ \ \ x < 1
\end{array} \right.\$
$= \left\{ \begin{array}{r}
2x - 3,\ \ x \geq 2 \\
1,\ \ \ \ 1 \leq x < 2 \\
3 - 2x,\ \ x < 1
\end{array} \right.\$
$f^{'}(x) = \left\{ \begin{array}{r}
2,\ \ x > 2 \\
0,\ \ \ 1 < x < 2 \\
- 1,\ \ \ \ x < 1
\end{array} \right.\$
Hence, except $x = 1$ and $x = 2,\ f(x)$ is differentiable everywhere in $R$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.