Question

Write the value of the derivative of f (x) = |x − 1| + |x − 3| at x = 2.

Answer 1

Given: $f\left(x\right)=\left|x-1\right|+\left|x-3\right|$

$\Rightarrow f\left(x\right)=\left\{\begin{array}{l}-(x-1)-(x-3),x<1\\ \begin{array}{l}x-1-(x-3),1\le x<3\\ (x-1)+(x-3),x\ge 3\end{array}\end{array}\right.$

$\Rightarrow f\left(x\right)=\left\{\begin{array}{l}-2x+4,x<1\\ \begin{array}{l}2,1\le x<3\\ 2x-4,x\ge 3\end{array}\end{array}\right.$

We check differentiability at x = 2

(LHD at x = 2)

$$\begin{gathered} \underset{x\to 2^{-}}{\lim }\frac{f\left(x\right)-f\left(2\right)}{x-2} \\ =\underset{h\to 0}{\lim }\frac{f(2-h)-f\left(2\right)}{2-h-2} \\ =\underset{h\to 0}{\lim }\frac{2-2}{-h} \\ =0 \end{gathered}$$