Mathematics

This problem asks you to find which even-order derivatives of $f$ are nonzero at $x=0$. The key insight is that $f(x)$ is an even function (both $x\tan^{-1}(x^2)$ and $x^4$ are even), so all odd derivatives vanish at the origin. For even derivatives, use the Taylor series: the coefficient of $x^{2m}$ in $f(x)$ tells you directly what $f^{2m}(0)$ is (up to a factorial). Expand each term carefully and identify which power of $x$ appears first with a nonzero coefficient.

This problem asks you to find which even-order derivatives of $f$ are nonzero at $x=0$. The key insight is that $f(x)$ is an even function (both $x\tan^{-1}(x^2)$ and $x^4$ are even), so all odd derivatives vanish at the origin. For even derivatives, use the Taylor series: the coefficient of $x^{2m}$ in $f(x)$ tells you directly what $f^{2m}(0)$ is (up to a factorial). Expand each term carefully and identify which power of $x$ appears first with a nonzero coefficient.