Q104 / 160 · AP EAMCET 2004 · Mathematics
If $f: R \rightarrow R$ is an even function having derivatives of all orders, then an odd function among the following is
- A$f^{\prime \prime}$
- Correct answerB$f^{\prime \prime \prime}$
- C$f^{\prime}+f^{\prime \prime}$
- D$f^{\prime \prime}+f^{\prime \prime \prime}$
Solution
Since, $f$ is an even function.
Let
$$
\begin{aligned}
f(x) & =\cos x \\
f^{\prime}(x) & =\sin x \\
f^{\prime \prime}(x) & =-\cos x \\
f^{\prime \prime \prime}(x) & =\sin x
\end{aligned}
$$
Since, $\sin x$ is an odd function. $\therefore$ In $f^{\prime \prime \prime}$ it is an odd function
Therefore option (2) is correct.
Hence, the correct answer is option: 'B'.