examday

/TMUA

The distinct cubic polynomials $P$ and $Q$ both have coefficient $1$ for $x^3$. Neither polynomial has a repeated real root, and

The curves $y=P(x)$ and $y=Q(x)$ intersect at exactly two distinct points.

Consider the following statements.

1. Exactly one of the two intersections lies on the $x$-axis.
2. The straight line joining the two intersections has positive gradient.
3. At each of the two intersections, the tangents to both curves have positive gradient.

Which of these statements must be true?