Question

At what point on the curve y=x+2cos x is the tangent horizontal in the interval [-2pi, 2pi]

Answer 1

$\bf y=x+2cos(x)\qquad \qquad [-2\pi ,2\pi ]\\\\ -----------------------------\\\\ \cfrac{dy}{dx}=1-2sin(x)\implies 0=1-2sin(x)\implies sin(x)=\cfrac{1}{2} \\\\\\ \measuredangle x=sin^{-1}\left( \frac{1}{2} \right)\implies \measuredangle x= \begin{cases} \frac{\pi }{6}\\\\ \frac{5\pi }{6} \end{cases}$

only in the 1st and 2nd quadrants, where sine is positive