Answer
Horizontal tangents at $(1, 0)$ and $(1, -4)$.
Vertical tangents at $(0, -2)$ and $(2, -2)$.
Work Step by Step
$\dfrac{d}{dx}(4x^2)+\dfrac{d}{dx}(y^2)-\dfrac{d}{dx}(8x)+\dfrac{d}{dx}(4y)+\dfrac{d}{dx}(4)=0\rightarrow$
$8x+\dfrac{dy}{dx}(2y)-8+\dfrac{dy}{dx}(4)=0\rightarrow$
$\dfrac{dy}{dx}=\dfrac{8x-8}{-2y-4}.$
Case I: Horizontal tangent $\rightarrow \dfrac{dy}{dx}=0\rightarrow8x-8=0\rightarrow x=1.$
Substituting in $x=1$ into the original equation gives:
$4(1^2)+y^2-8(1)+4y+4=y(y+4)=0\rightarrow$
$y=-4$ or $y=0\rightarrow$ horizontal tangents at $(1, 0)$ and $(1, -4)$.
Case II: Vertical tangents$\rightarrow\dfrac{dy}{dx}=$undefined$\rightarrow-2y-4=0\rightarrow y=-2.$
Substituting $y=-2$ into the original equation gives:
$4x^2+(-2)^2-8x+4(-2)+4=4x(x-2)\rightarrow x=0$ or $x=2\rightarrow$
vertical tangent at $(0, -2)$ and $(2, -2)$.
