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?
$f\left(x\right)=4x{\cos }^2\left(\frac{x}{2}\right)$ and $g\left(x\right)=8\cos \left(\frac{x}{2}\right)-3-2x\sin x$

Answer 1

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

$4{\cos }^2\left(\frac{x}{2}\right)-2x\sin x=8\cos \frac{x}{2}-3-2x\sin x$

$4{\cos }^2\frac{x}{2}-6\cos \frac{x}{2}-2\cos \frac{x}{2}+3=0$

$(2\cos \frac{x}{2}-1)(2\cos \frac{x}{2}+3)=0$

$\cos \frac{x}{2}=\frac{1}{2};\cos \frac{x}{2}\ne -\frac{3}{2}$

$\frac{x}{2}=2n\pi \pm \frac{\pi }{6}$

$x=4n\pi \pm \frac{\pi }{3}$