Differential Calculus
Differential Calculus
star_batch_jee_advanced_2025
Grade 12
Question:
Let $f(x) = x\tan^{-1}(x^2) + x^4$. Let $f^k(x)$ denotes kth derivative of $f(x)$ w.r.t. $x$, $k \in \mathbb{N}$. If $f^{2m}(0) \neq 0$, $m \in \mathbb{N}$, then $m$ equals to______.
Step-by-Step Solution
Key Concept: Decomposing $f(x)$ into odd and even functions and using the fact that odd function derivatives vanish at zero isolates which derivatives survive.
Let $g(x) = x\tan^{-1}(x^2)$, which is an odd function, so $g^{2n}(0) = 0$. Let $h(x) = x^4$. Then $f(x) = g(x) + h(x)$ and $f^{2n}(0) = g^{2n}(0) + h^{2n}(0) = h^{2n}(0)$. For the $2n$-th derivative of $x^4$ to vanish at $x=0$, we need $2n \geq 4 + 1$, which gives $2n = 4$, so $m = 2$.
Correct Answer: 3