Consider the curve given by $x ^ { 2 } + 4 y ^ { 2 } = 7 + 3 x y$.
(a) Show that $\frac { d y } { d x } = \frac { 3 y - 2 x } { 8 y - 3 x }$.
(b) Show that there is a point $P$ with $x$-coordinate 3 at which the line tangent to the curve at $P$ is horizontal. Find the $y$-coordinate of $P$.
(c) Find the value of $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ at the point $P$ found in part (b). Does the curve have a local maximum, a local minimum, or neither at the point $P$ ? Justify your answer.