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  • Let f: (–2, 2)  be defined by  <img alt="\mathrm{f}(\

Let f: (–2, 2) \rightarrow \mathbb{R} be defined by  \mathrm{f}(\mathrm{x})=\left\{\begin{array}{l} \mathrm{x}[\mathrm{x}] \quad,-2<\mathrm{x}<0 \\ (\mathrm{x}-1)[\mathrm{x}], 0 \leq \mathrm{x}<2 \end{array}\right.

  where [x] denotes the greatest integer function. If m and n respectively are the number of points in (–2, 2) at

which y = |f(x)| is not continuous and not differentiable, then m + n is equal to ______.

Option: 1

4


Option: 2

-


Option: 3

-


Option: 4

-


Answers (1)

best_answer

f(x)=\left\{\begin{array}{cc} -2 x, & -2<x<-1 \\ -x, & -1 \leq x<0 \\ 0, & 0 \leq x<1 \\ x-1, & 1 \leq x<2 \end{array}\right.

Clearly f(x) is discontinuous at x = –1 also non differentiable.
\therefore m = 1
Now for differentiability

f(x)=\left\{\begin{array}{cc} -2 x, & -2<x<-1 \\ -1, & -1 \leq x<0 \\ 0, & 0 \leq x<1 \\ -1, & 1 \leq x<2 \end{array}\right.

Clearly f(x) is non-differentiable at x = –1, 0, 1
Also, \left | f(x) \right | remains same.

\therefore n=3

\therefore m+n=4

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jitender.kumar

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