| Assertion [A]: \[f\left( x \right)\text{ }=\text{ }\left[ x \right]\]is not differentiable |
| Reason [R]: \[f\left( x \right)\text{ }=\text{ }\left[ x \right]\]is not continuous at\[x=0\]. |
A) Both A and R are individually true and R is the correct explanation of A.
B) Both A and R are individually true and R is not the correct explanation of A.
C) 'A' is true but 'R' is false
D) 'A' is false but 'R' is true
E) Both A and R are false.
Correct Answer: A
Solution :
Given Assertion: \[f\left( x \right)\text{ }=\text{ }\left[ x \right]\] Let us check continuity of fix) at x = n L.H. Lt \[=\underset{x\to {{n}^{1-}}}{\mathop{Lt}}\,[x]=\underset{h\to 0}{\mathop{Lt}}\,[n-h]=n-1\] R.H. Lt \[=\underset{x\to {{n}^{+}}}{\mathop{Lt}}\,[x]=\underset{h\to 0}{\mathop{Lt}}\,[n+h]=n\] Clearly L.H. Lt \[\ne \] R:H. Lt \[\therefore \,\,\,\,f\left( x \right)\]is not continuous at \[x\text{ }=\text{ }n\] \[\therefore \] Reason (R) is true Also we know that Every discontinuous function is not differentiable \[\therefore \,\,\,\,f\left( x \right)\]is also not differentiable \[\Rightarrow \]Given Assertion is true and given reason is correct explanation of Assertion Hence option [A] is the correct answer.
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