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Question
Write the number of points where f (x) = |x| + |x − 1| is continuous but not differentiable.
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Solution
Given:
\[f(x) = \left| x \right| + \left| x - 1 \right|\]
`⇒ f(x) = {(-x-(x-1), x<0),(x-(x-1),0le x<1),(x +(x-1),xge1):}`
`⇒ f(x) = {(-2x+1, x < 0 ),(1, 0le x <1),(2x -1, xge1):} `
When
Now,
(LHD at x = 0)
= 0
Hence
\[ = 0\]
\[ = \lim_{x \to 1} \frac{2(x - 1)}{x - 1} \]
\[ = 2\]
Therefore, 0,1 are the points where f(x) is continuous but not differentiable.
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