Calculus of a Single Variable, Chapter 10, 10.3, Section 10.3, Problem 31

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.