If ⨍ (x)=3x, -1≤x≤14-x, 1<x≤4, then

fx=3x ;-1≤x≤14-x ; 1<x≤4f1-=f1+=f1=3⇒f is continous f11-=limx→1-3x-3x-1=3 limx→1-3x-1-1x-1=3 log 3 f11+=limx→1+4-x-3x-1=-1 ∴f11-≠f11+⇒f is not differentiable at x=1

If ⨍ (x)=3x, -1≤x≤14-x, 1<x≤4, then

fx=3x ;-1≤x≤14-x ; 1<x≤4f1-=f1+=f1=3⇒f is continous f11-=limx→1-3x-3x-1=3 limx→1-3x-1-1x-1=3 log 3 f11+=limx→1+4-x-3x-1=-1 ∴f11-≠f11+⇒f is not differentiable at x=1

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If $⨍ (x)= \{\begin{array}{l} 3^{x}, - 1 \leq x \leq 1 \\ 4 - x, 1 < x \leq 4 \end{array},$ then

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$⨍ (x)$ is differentiable but not continuous at $x = 1$

$⨍ (x)$ is continuous but notdifferentiableat $x = 1$

$⨍ (x)$ is continuous as well as differentiable at $x = 1$

$⨍ (x)$ is neither continuous nor differentiable at $x = 1$

answer is B.

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Detailed Solution

$f (x) = \{\begin{cases} 3^{x}; - 1 \leq x \leq 1 \\ 4 - x; 1 < x \leq 4 \end{cases}$

$f (1^{-}) = f (1^{+}) = f (1) = 3 \Rightarrow f is continous f^{1} (1^{-}) = \lim_{x \to 1^{-}} \frac{3^{x} - 3}{x - 1} = 3 \lim_{x \to 1^{-}} \frac{3^{x - 1} - 1}{x - 1} = 3 \log 3 f^{1} (1^{+}) = \lim_{x \to 1^{+}} \frac{4 - x - 3}{x - 1} = - 1 ∴ f^{1} (1^{-}) \neq f^{1} (1^{+}) \Rightarrow f is not differentiable at x = 1$