Question

If $y=\left|\cos x\right|+\left|\sin x\right|$ then $\frac{dy}{dx}$ at $x=\frac{2\pi }{3}$ is:

• A. $\frac{1-\sqrt{3}}{2}$
• B. $0$
• C. $\frac{\sqrt{3}-1}{2}$
• D. None of these

Answer 1

The correct option is C $\frac{\sqrt{3}-1}{2}$

In the neighborhood of $x=\frac{2\pi }{3}$
$\Rightarrow y=-\cos x+\sin x$
$\Rightarrow \left(\frac{dy}{dx}\right)=\sin x+\cos x$
Thus at$x=\frac{2\pi }{3}$
$\frac{dy}{dx}=\frac{\sqrt{3}}{2}-\frac{1}{\sqrt{2}}=\frac{1}{2}(\sqrt{3}-1)$