Correct option is (2) \( 2 y^{\prime}+3 \pi^2 y=0\)
\( y=\sin ^3(\pi / 3 \cos g(x))\)
\(g(x)=\frac{\pi}{3 \sqrt{2}}\left(-4 x^3+5 x^2+1\right)^{3 / 2} \)
\(g(1)=2 \pi / 3\)
\(y^{\prime}=3 \sin ^2\left(\frac{\pi}{3} \cos g(x)\right) \times \cos \left(\frac{\pi}{3} \cos g(x)\right) \times \frac{\pi}{3}(-\sin g(x)) g^{\prime}(x)\)
\(y^{\prime}(1)=3 \sin ^2\left(-\frac{\pi}{6}\right) \cdot \cos \left(\frac{\pi}{6}\right) \cdot \frac{\pi}{3}\left(-\sin \frac{2 \pi}{3}\right) g^{\prime}(1)\)
\(g^{\prime}(x)=\frac{\pi}{3 \sqrt{2}}\left(-4 x^3+5 x^2+1\right)^{1 / 2}\left(-12 x^2+10 x\right)\)
\(g^{\prime}(1)=\frac{\pi}{2 \sqrt{2}}(\sqrt{2})(-2)=-\pi \)
\(y^{\prime}(1)=\frac{3}{4} \cdot \frac{\sqrt{3}}{2} \cdot \frac{\pi}{3}\left(\frac{-\sqrt{3}}{2}\right)(-\pi)=\frac{3 \pi^2}{16} \)
\(y(1)=\sin ^3(\pi / 3 \cos 2 \pi / 3)=-\frac{1}{8}\)
\( 2 y^{\prime}(1)+3 \pi^2 y(1)=0\)
