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Q.
All the points on the curve $y=\sqrt{x+\sin x}$ at which the tangent is parallel to $x$ - axis lie on a/an
Application of Derivatives
Solution:
$y=\sqrt{x+\sin x}$
Let at $P\left(x_{1}, y_{1}\right)$ tangent be parallel to $x$ -axis.
So $\left(\frac{dy}{dx}\right)_{\left(x_{1}, y_{1}\right)}=0$
$\Rightarrow \frac{1+\cos x_{1}}{2 \sqrt{x_{1}+\sin x_{1}}}=0$
$\Rightarrow 1+\cos x_{1}=0$
$\Rightarrow \cos x_{1}=-1 \Rightarrow x_{1}=\pi \Rightarrow \sin x_{1}=0\, \dots(i)$
Since $\left(x_{1}, y_{1}\right)$ lies on given curve,
$y_{1}^{2}=x_{1}+\sin x_{1} $
$\Rightarrow y_{1}^{2}=x_{1}$
$(\sin x_{1}=0$ from (i))
$\therefore $ Locus of $\left(x_{1}, y_{1}\right)$ is $y^{2}=x$
which is a parabola.