If $f: R \rightarrow R$ is an even function having derivatives of all orders, then an odd function among the…
If $f: R \rightarrow R$ is an even function having derivatives of all orders, then an odd function among the following is
$f^{\prime \prime}$
$f^{\prime \prime \prime}$
$f^{\prime}+f^{\prime \prime}$
$f^{\prime \prime}+f^{\prime \prime \prime}$
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