Question

Given two functions $f\left(x\right)$ and $g\left(x\right)$. For what $x$ does the equation $f^{\prime }\left(x\right)=g\left(x\right)$ hold true?
$f\left(x\right)={\sin }^{3}2x$ and $g\left(x\right)=4\cos 2x-5\sin 4x$

Answer 1

$f^{\prime }\left(x\right)=g\left(x\right)$

$6{\sin }^{2}2x\cos 2x=4\cos 2x-10\sin 2x\cos 2x$

$6{\sin }^{2}2x+10\sin 2x-4=0$

$6{\sin }^{2}2x+12\sin 2x-2\sin 2x-4=0$

$(6\sin 2x-2)(\sin 2x+2)=0$

$\sin 2x\ne -2but\sin 2x=\frac{1}{3}$

$x=\frac{n\pi }{2}+(-1{)}^m\frac{1}{2}\arcsin \left(\frac{1}{3}\right)$

$\cos 2x=0\Rightarrow x=(2n+1)\frac{\pi }{4}$