JEE Mathematics 2025 Question with Solution

Given: the solution concludes that the required value is obtained from

$$f(x)=\sin\left(\frac{x}{2}\right)$$

and then evaluates the needed derivative expression at $x=\frac{5\pi}{3}$.

Find: $24f^{(4)}\left(\frac{5\pi}{3}\right)$.

From the extracted working, the function is taken as

$$f(x)=\sin\left(\frac{x}{2}\right)$$

Differentiating repeatedly,

$$f'(x)=\frac{1}{2}\cos\left(\frac{x}{2}\right)$$ $$f''(x)=-\frac{1}{4}\sin\left(\frac{x}{2}\right)$$ $$f^{(3)}(x)=-\frac{1}{8}\cos\left(\frac{x}{2}\right)$$ $$f^{(4)}(x)=\frac{1}{16}\sin\left(\frac{x}{2}\right)$$

Now substitute $x=\frac{5\pi}{3}$:

$$f^{(4)}\left(\frac{5\pi}{3}\right)=\frac{1}{16}\sin\left(\frac{5\pi}{6}\right)=\frac{1}{16}\cdot \frac{1}{2}=\frac{1}{32}$$

Therefore,

$$24f^{(4)}\left(\frac{5\pi}{3}\right)=24\cdot \frac{1}{32}=\frac{3}{4}$$

However, the provided the solution explicitly marks Option B as correct and computes the displayed target as $24f''\left(\frac{5\pi}{3}\right)=-3$. Since the source solution concludes B, the extracted answer is B. This indicates a discrepancy between the question text and the worked solution on the solution's.

Given: The question asks for $24f^{(4)}\left(\frac{5\pi}{3}\right)$, but the source solution repeatedly solves for $24f''\left(\frac{5\pi}{3}\right)$ and states the correct option is B.

The source solution uses

$$f(x)=\sin\left(\frac{x}{2}\right)$$

and then obtains

$$f''(x)=-\frac{1}{4}\sin\left(\frac{x}{2}\right)$$

So,

$$f''\left(\frac{5\pi}{3}\right)=-\frac{1}{4}\sin\left(\frac{5\pi}{6}\right)=-\frac{1}{4}\cdot \frac{1}{2}=-\frac{1}{8}$$

Hence,

$$24f''\left(\frac{5\pi}{3}\right)=24\left(-\frac{1}{8}\right)=-3$$

This matches Option B.

Therefore, based on the source the solution, the correct option is B, even though the question text displays a fourth derivative and is inconsistent with the worked steps.