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If $f: R \rightarrow R$ is an even function having derivatives of all orders, then an odd function among the following is
Options:
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1347 Upvotes
Verified Answer
The correct answer is:
$f^{\prime \prime \prime}$
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.
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.
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