Consider the differential equation $\dfrac{dy}{dx} = \dfrac{3 - x}{y}$. (a) Let $y = f(x)$ be the particular solution to the given differential equation for $1 < x < 5$ such that the line $y = -2$ is tangent to the graph of $f$. Find the $x$-coordinate of the point of tangency, and determine whether $f$ has a local maximum, local minimum, or neither at this point. Justify your answer. (b) Let $y = g(x)$ be the particular solution to the given differential equation for $-2 < x < 8$, with the initial condition $g(6) = -4$. Find $y = g(x)$.
Consider the differential equation $\dfrac{dy}{dx} = \dfrac{3 - x}{y}$.
(a) Let $y = f(x)$ be the particular solution to the given differential equation for $1 < x < 5$ such that the line $y = -2$ is tangent to the graph of $f$. Find the $x$-coordinate of the point of tangency, and determine whether $f$ has a local maximum, local minimum, or neither at this point. Justify your answer.
(b) Let $y = g(x)$ be the particular solution to the given differential equation for $-2 < x < 8$, with the initial condition $g(6) = -4$. Find $y = g(x)$.