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

Answer 1

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

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

$x^2\sin x=0\Rightarrow x=n\pi$

$4x{\cos }^2\left(\frac{x}{2}\right)=x$

${\cos }^2\left(\frac{x}{2}\right)=\frac{1}{4}$

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

$x=n\pi ,2n\pi \pm \frac{\pi }{3}$