2024 TMJC P1 Q7

A curve \(C\) has equation \({{y}^{2}}=3{{x}^{3}}-x+1\).

1. Find the exact coordinates of all the stationary points.[4]
2. A particle moves along \(C\) to the stationary point \(S\), found in part (i) where both the \(x\)-coordinate and \(y\)-coordinate are positive. Given that its \(x\)-coordinate is increasing at a rate of \(\frac{1}{9}\) units per second, find the exact rate of change of the gradient of \(C\) when the particle is at point \(S\).[4]