Question

For what $x$ is the derivative of the function $f\left(x\right)$ equal to zero?
$f\left(x\right)=1-\sin (\pi +x)+2\cos (3\pi +x)/2)$

Answer 1

Given $f\left(x\right)=1-\sin (\pi +x)+2\cos \left(\frac{3\pi +x}{2}\right)$

$f^{\prime }\left(x\right)=-\cos (\pi +x)-\sin \left(\frac{3\pi +x}{2}\right)=0$

$\cos x+\sin \left(\frac{\pi +x}{2}\right)=0$

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

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

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

${\sin }^2\left(\frac{x}{2}\right)=0$

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

$x=2n\pi$