Consider the function,
[tex]f(x)=6\sin x+\frac{9}{8}[/tex]
The first derivative gives the slope (m) of the tangent of the curve,
[tex]\begin{gathered} m=f^{\prime}(x) \\ m=\frac{d}{dx}(6\sin x+\frac{9}{8}) \\ m=6\cos x+0 \\ m=6\cos x \end{gathered}[/tex]
The equation of the line is given as,
[tex]y-3\sqrt[]{3}x=\frac{7}{3}[/tex]
This can be written as,
[tex]y=3\sqrt[]{3}x+\frac{7}{3}[/tex]
Comparing with the slope-intercept form of the equation of a line, it can be concluded that the given line has a slope,
[tex]m^{\prime}=3\sqrt[]{3}[/tex]
Given that the tangent to the curve is parallel to this line, so their slopes must also be equal,
[tex]\begin{gathered} m=m^{\prime} \\ 6\cos x=3\sqrt[]{3} \\ \cos x=\frac{\sqrt[]{3}}{2} \\ \cos x=\cos (\frac{\pi}{6}) \end{gathered}[/tex]
Consider the formula,
[tex]\cos A=\cos B\Rightarrow A=2k\pi\pm B[/tex]
Applying the formula,
[tex]x=2k\pi\pm\frac{\pi}{6}[/tex]
Thus, the required values of 'x' are,
[tex]x=2k\pi\pm\frac{\pi}{6}[/tex]
Therefore, options 1st and 2nd are the correct choices.
