Question

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

Answer 1

Given:  

\[f(x) = \left| x - 1 \right| + \left| x - 3 \right|\]

`⇒ f(x) = {(-(x-1) - (x -3), x<1),(x - 1-(x -3), 1le x <3),((x-1)+(x-3), xge 3):}`

`⇒ f(x) = {(-2x +4, x<1),(2 , 1le x <3),(2x - 4,x ge 3):}`

We check differentiability at x = 2
(LHD at x = 2)

\[\lim_{x \to 2^-} \frac{f(x) - f(2)}{x - 2}\]
\[ = \lim_{h \to 0} \frac{f(2 - h) - f(2)}{2 - h - 2} \]
\[ = \lim_{h \to 0} \frac{2 - 2}{- h} \]
\[ = 0\]