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Let f(x) = \left\{\begin{aligned} \int_0^x(1+|2-t|) d t, & \text { if } x>4 \\ 4 x-4, & \text { if } x \leq 4 \end{aligned}\right.

Option: 1

f is continuous and differentiable at x = 4


Option: 2

 f is continuous but not differentiable at x = 4

 


Option: 3

 f is not differentiable at x = 4 

 


Option: 4

 none of these 

 


Answers (1)

best_answer

\begin{aligned} & f(x)=\int_0^2(4-t) d t+\int_2^x t d t=4+\frac{x^2}{2} \\ & \qquad \begin{cases}4+\frac{x^2}{2}, & x>4 \\ 4(x-1), & x \leq 4\end{cases} \\ & \therefore f(x)= \\ & \therefore \lim _{x \rightarrow 2^{+}} f(x)=\lim _{x \rightarrow 2^{-}} f(x)=\lim _{x \rightarrow 2} f(x)=12 \\ & \therefore \text { L.H.D. = R.H.D. }=4 . \end{aligned}

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