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Q.
The abscissas of points $P$ and $Q$ on the curve $y=e^{x}+e^{-x}$ such that tangents at $P$ and $Q$ make $60^{\circ}$ with the $x$ - axis are
Application of Derivatives
Solution:
$y=e^{x}+e^{-x}$ or $\frac{dy}{dx}=e^{x}-e^{-x}=\tan\, \theta$,
where $\theta$ is the angle of the tangent with the $x$ -axis.
For $\theta=60^{\circ}$, we have
$\tan \,60^{\circ}=e^{x}-e^{-x}$
or $e^{2 x}-\sqrt{3} e^{x}-1=0$
or $e^{x}=\frac{\sqrt{3} \pm \sqrt{7}}{2}$ or $x=\log _{e}\left(\frac{\sqrt{3}+\sqrt{7}}{2}\right)$