Question

The equation of the normal to the curve $y=x+\sin x\cos x$ at $x=\pi /2$ is

• A. $x=2$
• B. $x=\pi$
• C. $x+\pi =0$
• D. $2x=\pi$

Answer 1

The correct option is D $2x=\pi$

$y=x+\sin x\cos x=x+\frac{\sin \left(2x\right)}{2}$
$y^{\prime }=1+\cos \left(2x\right)$
Substituting $x=\pi /2$
$y^{\prime }\left(\pi /2\right)=1+\cos \left(\pi \right)=0$
Therefore, slope of tangent is 0. This means that the normal is a vertical line of the $x=c$. Since the point has x-coordinate $\pi /2$, the equation of the line is:
$x=\pi /2⟹2x=\pi$