JEE MAIN - Mathematics (2023 - 10th April Morning Shift - No. 16)
Let $$f:( - 2,2) \to R$$ be defined by $$f(x) = \left\{ {\matrix{
{x[x],} & { - 2 < x < 0} \cr
{(x - 1)[x],} & {0 \le x \le 2} \cr
} } \right.$$ 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 ____________.
Answer
4
Explanation
Given function is $f(x)=\left\{\begin{array}{cc}x[x], & -2 < x < 0 \\ (x-1)[x], & 0 \leq x<2\end{array}\right.$
When $[x]$ is denotes greatest integer function
_10th_April_Morning_Shift_en_16_1.png)
Clearly, $|f(x)|$ remains same.
Given that, $m$ and $n$ respectively are the number points in $(-2,2)$ at which $y=|f(x)|$ is not continuous and not differentiable
So, $m=1$ where $y=|f(x)|$ not continuous
and $n=3$ where $|f(x)|$ is not differentiable.
Thus, $m+n=4$
When $[x]$ is denotes greatest integer function
Clearly, $|f(x)|$ remains same.
Given that, $m$ and $n$ respectively are the number points in $(-2,2)$ at which $y=|f(x)|$ is not continuous and not differentiable
So, $m=1$ where $y=|f(x)|$ not continuous
and $n=3$ where $|f(x)|$ is not differentiable.
Thus, $m+n=4$
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