Question

Find the points at which the tangent to the curve $y=x-0.5sin2x-0.5cos2x+16cosx$ is parallel to the abscissa axis.

Answer 1

$y=x-0.5\sin 2x-0.5\cos 2x+16\cos x$

absaissa axis $\to x-axis$

$slope=0$

Tangent equation $:\frac{dy}{dx}|(x,y)=\frac{y-y_1}{x-x_1}$

Points at which is $0\to$ Tangent are parallel to x-axis

$\frac{dy}{dx}=\frac{d}{dx}[x-0.5(\sin 2x+\cos 2x)+16\cos x]$

$0=1-0.5(\cos 2x-\sin 2x).2-16\cos x$

$0=1-\sqrt{2}\left(\sin \right(\frac{\pi }{4})-2x)-16\sin x$

$0=1-({\cos }^{2}x-{\sin }^{2}x-2\sin x\cos x)-16\sin x$

$=1-({\cos }^{2}x-{\sin }^{2}x+2\sin x\cos x)-16\sin x$

$0=25si{n}^{2}x+2\sin x\cos x-16\sin x$

For zero $:\sin x=0$

$x=x\pi$ where $n=0,1,\mathrm{2....}$

$y=n\pi -0.5+16(-1{)}^n$

$(n\pi ,n\pi -0.5+16(-1{)}^n)$ where $n=0,1,\mathrm{2.....}$

are the points where the curve is parallel to absaissa axis.