Consider the function $y = f(x)$ whose curve is given by the equation $2y^{2} - 6 = y\sin x$ for $y > 0$. (a) Show that $\frac{dy}{dx} = \frac{y\cos x}{4y - \sin x}$. (b) Write an equation for the line tangent to the curve at the point $(0, \sqrt{3})$. (c) For $0 \leq x \leq \pi$ and $y > 0$, find the coordinates of the point where the line tangent to the curve is horizontal. (d) Determine whether $f$ has a relative minimum, a relative maximum, or neither at the point found in part (c). Justify your answer.
Consider the function $y = f(x)$ whose curve is given by the equation $2y^{2} - 6 = y\sin x$ for $y > 0$.\\
(a) Show that $\frac{dy}{dx} = \frac{y\cos x}{4y - \sin x}$.\\
(b) Write an equation for the line tangent to the curve at the point $(0, \sqrt{3})$.\\
(c) For $0 \leq x \leq \pi$ and $y > 0$, find the coordinates of the point where the line tangent to the curve is horizontal.\\
(d) Determine whether $f$ has a relative minimum, a relative maximum, or neither at the point found in part (c). Justify your answer.