Problem 2. Consider the parametric curve
\(x = f(t) = t^2\), \(y = g(t) = t^3 - t\), \( -\infty < t < \infty \).
Prove that this curve intersects itself at precisely one point in the xy-plane. Find the parameters \(t_1\) and \(t_2\)
such that \(t_1 < t_2\), \(f(t_1) = f(t_2)\) and \(g(t_1) = g(t_2)\). At the intersection \((x, y) = (f(t_1), g(t_1)) = (f(t_2), g(t_2))\),
there are two tangent lines to this parametric curve in the xy-plane. Find the slopes of these two tangent
lines. Find the equations of the two tangent lines. Set up the definite integral that equals the length of the
parametric curve from \(t = t_1\) to \(t = t_2\). Do not evaluate this integral. The parametric curve forms a closed
loop for \(t_1 \le t \le t_2\).