Let y = f ( x ) = sin 3 ( 3 ( cos ( 3 2 ( - 4 x 3 + 5 x 2 + 1 ) ) 3 2 ) ) . Then, at x = 1 , [2023]

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Solution

Q.
$Let y = f (x) = \sin^{3} (\frac{π}{3} (\cos (\frac{π}{3 \sqrt{2}} {(- 4 x^{3} + 5 x^{2} + 1)}^{\frac{3}{2}}))) .$ $Then, at x = 1,$ [2023]

1 $2 y' + \sqrt{3} π^{2} y = 0$

2 $\sqrt{2} y' - 3 π^{2} y = 0$

3 $2 y' + 3 π^{2} y = 0$

4 $y' + 3 π^{2} y = 0$

Ans.
(3)

$Let - 4 x^{3} + 5 x^{2} + 1 = t, then$

$y = \sin^{3} (\frac{π}{3} (\cos (\frac{π}{3 \sqrt{2}} t^{3 / 2})))$

$\frac{d t}{d x} = - 12 x^{2} + 10 x$

$∴$ $\frac{d y}{d x} = 3 \sin^{2} (\frac{π}{3} (\cos (\frac{π}{3 \sqrt{2}} t^{3 / 2}))) \cdot \cos (\frac{π}{3} \cos (\frac{π}{3 \sqrt{2}} t^{3 / 2})) \cdot \frac{π}{3} (- \sin (\frac{π}{3 \sqrt{2}} t^{3 / 2}))$

$\frac{π}{3 \sqrt{2}} \cdot \frac{3}{2} t^{1 / 2} \cdot \frac{d t}{d x}$

$\Rightarrow \frac{d y}{d x} = - 3 \sin^{2} (\frac{π}{3} (\cos (\frac{π}{3 \sqrt{2}} {(- 4 x^{3} + 5 x^{2} + 1)}^{3 / 2}))$ $\cdot \cos (\frac{π}{3} \cos (\frac{π}{3 \sqrt{2}} {(- 4 x^{3} + 5 x^{2} + 1)}^{3 / 2})) \cdot \frac{π}{3} \sin (\frac{π}{3 \sqrt{2}} {(- 4 x^{3} + 5 x^{2} + 1)}^{3 / 2})$ $\cdot \frac{π}{2 \sqrt{2}} {(- 4 x^{3} + 5 x^{2} + 1)}^{1 / 2} (- 12 x^{2} + 10 x)$

$At x = 1, t = - 4 + 5 + 1 = 2$

${\frac{d y}{d x}}_{(x = 1)} = - 3 \sin^{2} (\frac{π}{3} \cos (\frac{π}{3 \sqrt{2}} 2^{3 / 2})) \cdot \cos (\frac{π}{3} \cos (\frac{π}{3 \sqrt{2}} 2^{3 / 2}))$ $\cdot \frac{π}{3} \sin (\frac{π}{3 \sqrt{2}} 2^{3 / 2}) \cdot \frac{π}{2 \sqrt{2}} 2^{1 / 2} \cdot (- 2)$

$= π^{2} \sin^{2} (\frac{π}{3} (\cos \frac{2 π}{3})) \cdot \cos (\frac{π}{3} \cos \frac{2 π}{3}) \cdot \sin (\frac{2 π}{3})$

$= π^{2} \sin^{2} (- \frac{π}{3} \cdot \frac{1}{2}) \cdot \cos (- \frac{π}{3} \cdot \frac{1}{2}) \frac{\sqrt{3}}{2}$

$= \frac{\sqrt{3}}{2} π^{2} \cdot \frac{1}{4} \cdot \frac{\sqrt{3}}{2} = \frac{3}{16} π^{2}$

$at x = 1, y = \sin^{3} (\frac{π}{3} (\cos \frac{π}{3 \sqrt{2}} {(2)}^{3 / 2})) = \sin^{3} (\frac{π}{3} (\cos \frac{2 π}{3}))$

$= \sin^{3} (- \frac{π}{3} \times \frac{1}{2}) = - \frac{1}{8}$

$\Rightarrow 2 y' + 3 π^{2} y = 2 \times \frac{3}{16} π^{2} + 3 π^{2} \times (- \frac{1}{8}) = 0$

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