Question

Let $y=f(x)=\sin ^3\left(\frac{\pi}{3}\left(\cos \left(\frac{\pi}{3 \sqrt{2}}\left(-4 x^3+5 x^2+1\right)^{\frac{3}{2}}\right)\right)\right)$ Then, at $x=1$

• A. $2 y^{\prime}+\sqrt{3} \pi^2 y=0$
• B. $\sqrt{2} y^{\prime}-3 \pi^2 y=0$
• C. $2 y^{\prime}+3 \pi^2 y=0$
• D. $y^{\prime}+3 \pi^2 y=0$

Answer 1

Answer: (C)

$2 y^{\prime}+3 \pi^2 y=0$

Answer 2

y=sin3(π/3cosg(x))
g(x)=32​π​(−4x3+5x2+1)3/2
g(1)=2π/3
y′=3sin2(3π​cosg(x))×cos(3π​cosg(x))×3π​(−sing(x))g′(x)
y′(1)=3sin2(−6π​)⋅cos(6π​)⋅3π​(−sin32π​)g′(1)
g′(x)=32​π​(−4x3+5x2+1)1/2(−12x2+10x)
g′(1)=22​π​(2​)(−2)=−π
y′(1)=43​⋅23​​⋅3π​(2−3​​)(−π)=163π2​
y(1)=sin3(π/3cos2π/3)=−81​
2y′(1)+3π2y(1)=0