Question

Assertion :Let $f\left(x\right)=x$ for $x\in Q,f\left(x\right)=x^2+x$ for irrational $x$, then $f$ is differentiable at $x=0$ Reason: $f^{\prime }\left(0\right)=1$

• 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 A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

$f^{\prime }\left(x\right)=\{\begin{array}{c}1forx\in Q\\ 2x+1forx\in Q\end{array}{f}^{\prime }\left(0^{+}\right)=2\left(0\right)+1=f^{\prime }\left(0^{-}\right)$

Since $0^{+}$ and $0^{-}$ will be considered irrational.

$f^{\prime }\left(0\right)=1$ Zero is Rational $\Rightarrow f$ is differentiable at $x=0$

and, $f^{\prime }\left(0\right)=1$

Hence, assertion and reason are correct and reason is the correct explanation for assertion.