If f: R rightarrow R is an even function whic | vedclass

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(C) Given that $f(x)$ is an even function,we have $f(x) = f(-x)$ for all $x \in R$.Differentiating both sides with respect to $x$,we get $f^{\prime}(x

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$1$

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(C) Given that $f(x)$ is an even function,we have $f(x) = f(-x)$ for all $x \in R$.Differentiating both sides with respect to $x$,we get $f^{\prime}(x) = -f^{\prime}(-x)$.Differentiating again with respect to $x$,we get $f^{\prime \prime}(x) = f^{\prime \prime}(-x)$.This shows that the second derivative $f^{\prime \prime}(x)$ is also an even function.Since $f^{\prime \prime}(x)$ is an even function,$f^{\prime \prime}(-\pi) = f^{\prime \prime}(\pi)$.Given that $f^{\prime \prime}(\pi) = 1$,it follows that $f^{\prime \prime}(-\pi) = 1$.

If $f: R \rightarrow R$ is an even function which is twice differentiable on $R$ and $f^{\prime \prime}(\pi)=1$,then $f^{\prime \prime}(-\pi)$ is equal to

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