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Question
Find whether the function is differentiable at x = 1 and x = 2
Solution
\[f\left( x \right) = \begin{cases}x & x \leq 1 \\ \begin{array} 22 - x \\ - 2 + 3x - x^2\end{array} & \begin{array}11 \leq x \leq 2 \\ x > 2\end{array}\end{cases}\]
\[ \Rightarrow f'\left( x \right) = \begin{cases}1 & x \leq 1 \\ \begin{array} -- 1 \\ 3 - 2x\end{array} & \begin{array}11 \leq x \leq 2 \\ x > 2\end{array}\end{cases}\]
\[\text { Now }, \]
\[\text { LHL } = \lim_{x \to 1^-} f'\left( x \right) = \lim_{x \to 1^-} 1 = 1\]
\[\text { RHL } = \lim_{x \to 1^+} f'\left( x \right) = \lim_{x \to 1^+} - 1 = - 1\]
\[\text { Since , at x } = 1, \text { LHL} \neq \text { RHL }\]
\[\text { Hence }, f\left( x \right) \text { is not differentiable at } x = 1\]
\[\text { Again }, \]
\[\text { LHL }= \lim_{x \to 2^-} f'\left( x \right) = \lim_{x \to 2^-} - 1 = - 1\]
\[\text { RHL }= \lim_{x \to 2^+} f'\left( x \right) = \lim_{x \to 2^+} 3 - 2x = 3 - 4 = - 1\]
\[\text { Since , at x = 2, LHL = RHL}\]
\[\text { Hence,} f\left( x \right) \text { is differentiable at } x = 2\]
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