Let be a twice differentiable function such that , for all .... | JEE Main 2024 Differentiation

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Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a twice differentiable function such that $(\sin x \cos y)(f(2 x+2 y)-f(2 x-2 y))=(\cos x \sin y)(f(2 x+2 y)+f(2 x-2 y))$ , for all $x, y \in \mathbf{R}$ . If $f^{\prime}(0)=\frac{1}{2}$ , then the value of $24 f^{\prime \prime}\left(\frac{5 \pi}{3}\right)$ is :

$\begin{aligned} & \sin (x-y) f(2 x+2 y)=f(2 x-2 y) \sin (x+y) \\ & \frac{f(2 x+2 y)}{\sin (x+y)}=\frac{f(2 x-2 y)}{\sin (x-y)}=k(\text { say }) \\ & f(2 x+2 y)=k \sin (x+y) \\ & f(2 x)=5 \sin x \quad(\because y=0) \\ & f(x)=k \sin \frac{x}{2} \\ & f^{\prime}(x)=\frac{k}{2} \cos \frac{x}{2} \\ & f^{\prime}(0)=\frac{1}{2} \Rightarrow k=1 \\ & f(x)=\sin \frac{x}{2} \Rightarrow f^{\prime}(x)=\frac{1}{2} \cos \frac{x}{2} \\ & f^{\prime \prime}(x)=-\frac{1}{4} \sin \frac{x}{2} \\ & 24 f^{\prime \prime}\left(\frac{5 \pi}{3}\right)=-3 \end{aligned}$

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