Consider the closed curve in the $x y$-plane given by $x ^ { 2 } + 2 x + y ^ { 4 } + 4 y = 5$. (a) Show that $\frac { d y } { d x } = \frac { - ( x + 1 ) } { 2 \left( y ^ { 3 } + 1 \right) }$. (b) Write an equation for the line tangent to the curve at the point $( - 2,1 )$. (c) Find the coordinates of the two points on the curve where the line tangent to the curve is vertical. (d) Is it possible for this curve to have a horizontal tangent at points where it intersects the $x$-axis? Explain your reasoning.
Consider the closed curve in the $x y$-plane given by $x ^ { 2 } + 2 x + y ^ { 4 } + 4 y = 5$.
(a) Show that $\frac { d y } { d x } = \frac { - ( x + 1 ) } { 2 \left( y ^ { 3 } + 1 \right) }$.
(b) Write an equation for the line tangent to the curve at the point $( - 2,1 )$.
(c) Find the coordinates of the two points on the curve where the line tangent to the curve is vertical.
(d) Is it possible for this curve to have a horizontal tangent at points where it intersects the $x$-axis? Explain your reasoning.