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Assertion :If $$f\left ( x \right )=\left [ x \right ]\left ( \sin x-\cos x+2 \right )$$ (where [.] denotes the greatest integer function) then $${f}'\left ( x \right )=\left [ x \right ]\left ( \cos x+\sin x \right )$$ for $$\forall x\epsilon Z$$ Reason: $${f}'\left ( x \right )$$ does not exist for any $$x\epsilon $$ integer


A
Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
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B
Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
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C
Assertion is correct but Reason is incorrect
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D
Both Assertion and Reason are incorrect
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Solution

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
$$f\left ( x \right )=\left [ x \right ]\left ( \sin x-\cos x+2 \right )$$
$$\therefore $$   $${f}'\left ( x \right )=\left [ x \right ]\left ( \cos x+\sin x \right )$$ ($$\therefore $$ Assertion A is true)
Again let $$x=a\epsilon I$$ then
$${f}'\left ( a \right )^{+}\neq {f}'\left ( a \right )^{-}$$
$$\therefore $$   $${f}'\left ( a \right )$$ does not exist

Mathematics

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