Question

Assertion :The function $f\left(x\right)={\int }_0^x\sqrt{1+t^2}dt$ is an odd function and $g\left(x\right)=f^{\prime }\left(x\right)$ is an even function. Reason: For a differentiable function $f\left(x\right)$ if $f^{\prime \prime }\left(x\right)$ is an even function, then $f\left(x\right)$ is an odd function.

• 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 C Assertion is correct but Reason is incorrect

$f\left(x\right)={\int }_0^x\sqrt{1+t^2}dt$
$\Rightarrow f^{\prime }\left(x\right)=\sqrt{1+x^2}$
Thus, $g\left(x\right)=f^{\prime }\left(x\right)$ is an even function.
So, assertion is true.
Let $f^{\prime \prime }\left(x\right)=x^2$
$f^{\prime }\left(x\right)=\frac{x^3}{3}+c$
$f\left(x\right)=\frac{x^4}{12}+cx$
which is an odd function.Therefore, reason is incorrect.
Hence, option 'C' is correct.