Let ⨍ x=xx 2 then ⨍ (x) is

fx=x : x 2 limx→1-fx=limx→1+fx=1limx→2-fx=limx→1+fx=0 f is a continuous function f11-=1, f11+=-1⇒f is not differentiable at x=1 f12-=-1, f12+=-1⇒f is differentiable at x=2

Let ⨍ x=xx 2 then ⨍ (x) is

fx=x : x 2 limx→1-fx=limx→1+fx=1limx→2-fx=limx→1+fx=0 f is a continuous function f11-=1, f11+=-1⇒f is not differentiable at x=1 f12-=-1, f12+=-1⇒f is differentiable at x=2

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Let $⨍ (x) = \{\begin{matrix} x & x < 1 \\ 2 - x & 1 \leq x \leq 2 \\ - 2 + 3 x - x^{2} & x > 2 \end{matrix}$ then $⨍ (x)$ is

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differentiable at $x = 1$

differentiable at $x = 2$

differentiable at $x = 1$ and $x = 2$

not differentiable at $x = 0$

answer is B.

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Detailed Solution

$f (x) = \{\begin{cases} x : x < 1 \\ 2 - x : 1 \leq x \leq 2 \\ - 2 + 3 x - x^{2} : x > 2 \end{cases} \lim_{x \to 1^{-}} f (x) = \lim_{x \to 1^{+}} f (x) = 1$

$\lim_{x \to 2^{-}} f (x) = \lim_{x \to 1^{+}} f (x) = 0 f is a continuous function f^{1} (1^{-}) = 1, f^{1} (1^{+}) = - 1 \Rightarrow f is not differentiable at x = 1 f^{1} (2^{-}) = - 1, f^{1} (2^{+}) = - 1 \Rightarrow f is differentiable at x = 2$