Let x=t^3-3t
y=t^2-t
Where is the tangent parallel to the x-axis? y-axis? line y=x?
The curve intersects itself once. Find this point.
Thanks.
Horizontal tangents occur when $\displaystyle \frac{dy}{dt}=0$
Vertical tangents occur when $\displaystyle \frac{dx}{dt}=0$
to find the derivative use the formula
$\displaystyle \frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}= \frac{2t-1}{3t^2-3}$
For the last here is a hint.
For what values of t will the function have the same y-coordinate.
Find these and compare those times with the x -coordinate.