Limit Continuity And Differentiability Ques 37
- In the following, $[x]$ denotes the greatest integer less than or equal to $x$.
| Column I | Column II | ||
|---|---|---|---|
| A. | $x|x|$ | p. | continuous in $(-1,1)$ |
| B. | $\sqrt{|x|}$ | $\mathrm{q}$. | differentiable in $(-1,1)$ |
| C. | $x+[x]$ | $\mathrm{r}$. | strictly increasing $(-1,1)$ |
| D. | $|x-1|+|x+1|$ in $(-1,1)$ |
s. | not differentiable atleast at one point in $(-1,1)$ |
$(2007,6 \mathrm{M})$
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Answer:
Correct Answer: 37.$(\mathrm{A}) \rightarrow \mathrm{p}, \mathrm{q}, \mathrm{r}, \mathrm{s} ;(\mathrm{B}) \rightarrow \mathrm{p}, \mathrm{s} ;(\mathrm{C}) \rightarrow \mathrm{r}, \mathrm{s} ; (\mathrm{D}) \rightarrow \mathrm{p}, \mathrm{s}$
Solution: A. $x|x|$ is continuous, differentiable and strictly increasing in $(-1,1)$.
B. $\sqrt{|x|}$ is continuous in $(-1,1)$ and not differentiable at $x=0$.
C. $x+[x]$ is strictly increasing in $(-1,1)$ and discontinuous at $x=0$ $\Rightarrow$ not differentiable at $x=0$.
D. $|x-1|+|x+1|=2$ in $(-1,1)$ $\Rightarrow$ The function is continuous and differentiable in $(-1,1)$.
BETA
