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Question
Write the value of the derivative of f (x) = |x − 1| + |x − 3| at x = 2.
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Solution
Given:
`⇒ f(x) = {(-(x-1) - (x -3), x<1),(x - 1-(x -3), 1le x <3),((x-1)+(x-3), xge 3):}`
We check differentiability at x = 2
(LHD at x = 2)
\[\lim_{x \to 2^-} \frac{f(x) - f(2)}{x - 2}\]
\[ = \lim_{h \to 0} \frac{f(2 - h) - f(2)}{2 - h - 2} \]
\[ = \lim_{h \to 0} \frac{2 - 2}{- h} \]
\[ = 0\]
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