<s>[INST] Number of solution of equation $16 \sin^3x=14+(\sin x+7)^{\frac{1}{3}}$ in $[0,4\pi]$. Number of solutions of equation $16 \sin^3x=14+(\sin x+7)^{\dfrac{1}{3}}$ in $[0,4\pi]$.
My thinking:

*

*Directly satisfy $\sin x=1$.


*I thought of doing $f(x)=f^{-1}(x)\;$ but couldn't proceed further because I can't think of a function on the left side whose inverse is the right side.
 [/INST] Let $y=\sin x$. Then $$16(y^3-1)=\sqrt[3]{y+7}-2$$
$$16(y^3-1)\left(\left(\sqrt[3]{y+7}\right)^2+2\sqrt[3]{y+7}+4\right)=y-1$$
Then $y=1$ or $$16(y^2+y+1)\left(\left(\sqrt[3]{y+7}\right)^2+2\sqrt[3]{y+7}+4\right)=1$$
$$4(4y^2+4y+4)\left(\left(\sqrt[3]{y+7}\right)^2+2\sqrt[3]{y+7}+4\right)=1$$
$$4\left((2y+1)^2+3\right)\left(\left(\sqrt[3]{y+7}+1\right)^2+3\right)=1$$
But $$4\left((2y+1)^2+3\right)\left(\left(\sqrt[3]{y+7}+1\right)^2+3\right) \ge 4 \cdot 3 \cdot 3= 36 >1$$.
Only solution $y=1$
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