Question

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

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

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

$f\left(x\right)=\left[x\right](\sin x-\cos x+2)$
$\therefore$ $f^{\prime }\left(x\right)=\left[x\right](\cos x+\sin x)$ ($\therefore$ Assertion A is true)
Again let $x=aϵI$ then
$f^{\prime }{\left(a\right)}^{+}\ne f^{\prime }{\left(a\right)}^{-}$
$\therefore$ $f^{\prime }\left(a\right)$ does not exist