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  • Let  denote the greatest integer function and <img alt="f(x)=\max \{1+x+[x], 2+x, x+2[x]\}, 0 \leq x \leq 2" src=

Let [x] denote the greatest integer function and f(x)=\max \{1+x+[x], 2+x, x+2[x]\}, 0 \leq x \leq 2 Let m be the number of points in [0,2], where f is not continuous and n be the number of points in (0,2), where f is not differentiable. Then (\mathrm{m}+\mathrm{n})^2+2 is equal to

Option: 1

6


Option: 2

3


Option: 3

2


Option: 4

11


Answers (1)

best_answer

$$ \begin{aligned} & \text { Let } g(x)=1+x+[x]=\left\{\begin{array}{cc} 1+x ; & x \in[0,1) \\ 2+x ; & x \in[1,2) \\ 5 ; & x=2 \end{array}\right. \\ & \lambda(x)=x+2[x]=\left\{\begin{array}{cc} x ; & x \in[0,1) \\ x+2 ; & x \in[1,2) \\ 6 ; & x=2 \end{array}\right. \\ & r(x)=2+x \\ & f(x)=\left\{\begin{array}{cc} 2+x ; & x \in[0,2) \\ 6 ; & x=2 \end{array}\right.\\ &\mathrm{f}(\mathrm{x}) \text{ is discontinuous only at } \mathrm{x}=2 \Rightarrow \mathrm{m}=1\\ &\mathrm{f}(\mathrm{x}) \text { is differentiable in } (0,2) \Rightarrow \\ &\mathrm{n}=0 (m+n)^2+2=3\\ \end{aligned}

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SANGALDEEP SINGH

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