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Question Number 57959 by rahul 19 last updated on 15/Apr/19

Answered by MJS last updated on 16/Apr/19

$${3\sin }^{-1}x=y\Rightarrow x=\sin \frac{y}{3}$$$$\mathrm{with}x\in [-1;1]\wedge y\in [-\frac{3\pi }{2};\frac{3\pi }{2}]$$$${3\sin }^{-1}x\mathrm{increases}\mathrm{for}-1⩽x⩽1$$$$$$$${\sin }^{-1}(-4x^3+3x)=y\Rightarrow -4x^3+3x=\sin y$$$$\mathrm{with}(-4x^3+3x)\in [-1;1]\wedge y\in [-\frac{\pi }{2};\frac{\pi }{2}]$$$$(-4x^3+3x)\in [-1;1]\Rightarrow x\in [-1;1]$$$$\mathrm{but}-4x^3+3x\mathrm{has}\mathrm{a}\mathrm{minimum}\mathrm{at}\left(\begin{array}{c}-\frac{1}{2}\\ -1\end{array}\right)$$$$\mathrm{and}\mathrm{a}\mathrm{maximum}\mathrm{at}\left(\begin{array}{c}\frac{1}{2}\\ 1\end{array}\right)\Rightarrow$$$$\Rightarrow {\sin }^{-1}(-4x^3+3x)\mathrm{decreases}\mathrm{for}-1⩽x<-\frac{1}{2}$$$$\mathrm{and}\frac{1}{2}<x⩽1\mathrm{and}\mathrm{increases}\mathrm{for}-\frac{1}{2}<x<\frac{1}{2}$$$$$$$$\Rightarrow \mathrm{answer}\mathrm{is}\left(\mathrm{a}\right)$$

Commented by MJS last updated on 16/Apr/19

$$\mathrm{solving}-4x^3+3x=\sin y$$$$x^3-\frac{3}{4}x+\frac{1}{4}\sin y=0$$$$D=\frac{p^3}{27}+\frac{q^2}{4}=\frac{{(-\frac{3}{4})}^3}{27}+\frac{{\left(\frac{1}{4}\sin y\right)}^2}{4}=\frac{-1+{\sin }^{2}y}{64}=$$$$=-\frac{{\cos }^{2}y}{64}⩽0\Rightarrow 3\mathrm{real}\mathrm{solutions}\Rightarrow \mathrm{we}\mathrm{need}$$$$\mathrm{the}\mathrm{trigonometric}\mathrm{method},\mathrm{which}\mathrm{gives}$$$$x_1=\sin \frac{y}{3}$$$$x_2=\frac{1}{2}(\sqrt{3}\cos \frac{y}{3}-\sin \frac{y}{3})$$$$x_3=-\frac{1}{2}(\sqrt{3}\cos \frac{y}{3}+\sin \frac{y}{3})$$$$\Rightarrow {3\sin }^{-1}x={\sin }^{-1}(-4x^3+3x)\mathrm{in}\mathrm{certain}\mathrm{cases}$$

Commented by rahul 19 last updated on 17/Apr/19

$$Thankyousir.$$

Commented by MJS last updated on 17/Apr/19

$${\mathrm{you}}^{\prime }\mathrm{re}\mathrm{welcome}.\mathrm{I}\mathrm{hope}\mathrm{my}\mathrm{answer}\mathrm{is}\mathrm{helpful}$$$${\mathrm{I}}^{\prime }\mathrm{m}\mathrm{not}100\mathrm{\%}\mathrm{sure}\mathrm{if}\mathrm{this}\mathrm{is}\mathrm{a}\mathrm{legal}\mathrm{proof}...$$

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