Oxford University MAT 2021 Solved Papers

The tangent at a is y=(3a2−3)(x−a)+(a3−3a), which passes through (2,0) if and only if 0=(3a2−3)(2−a)+(a3−3a). This simplifies to 2a3−6a2+6=0. The left-hand side is a cubic in a and we'd like to know how many roots it has. The turning points of 2a3−6a2+6 are at a=0 and a=2, where the value of the cubic is 6 and −2 respectively. So this cubic starts negative, rises to a positive local maximum, then decreases to a negative local minimum before rising again. There are therefore three roots for this cubic, so three values of a for which the tangent to the original cubic passes through the point (2,0).