Parametric curve (x(t),y(t)) has a horizontal tangent if its slope `dy/dx` is zero, i.e when `dy/dt=0` and `dx/dt!=0`
Curve has a vertical tangent line, if its slope approaches infinity i.e `dx/dt=0`
and `dy/dt!=0`
Given parametric equations are:
`x=t+4`
`y=t^3-3t`
`dx/dt=1`
`dy/dt=3t^2-3`
For Horizontal tangents,
`dy/dt=0`
`3t^2-3=0`
`=>3t^2=3`
`=>t^2=1`
`=>t=+-1`
Corresponding points on the curve can be found by plugging in the values of t in the equations,
For t=1,
`x_1=1+4=5`
`y_1=1^3-3(1)=-2`
For t=-1,
`x_2=-1+4=3`
`y_2=(-1)^3-3(-1)=2`
Horizontal tangents are at the points (5,-2) and (3,2)
For vertical tangents,
`dx/dt=0`
However `dx/dt=1!=0`
So the curve has no vertical tangents.