TMUA Stationary Points

A stationary point of a curve is a point where the gradient is zero. On TMUA Paper 1, the typical pattern is: find the stationary points of a given cubic or quartic, classify them, and answer some question about their $y$-coordinates, distances, or a parameter that makes them coincide.

• Locating stationary points. Solve $y' = 0$. Substitute back into $y$ for the $y$-coordinates.
• Classifying. $y'' > 0$ at a stationary point: local minimum. $y'' < 0$: local maximum. $y'' = 0$: inconclusive, inspect $y'$ either side.
• Parameter conditions. ”$y$ has a stationary point at $x = 2$” is shorthand for $y'(2) = 0$.
• Counting stationary points. A cubic has two when its derivative has two real roots, one (inflection) when the derivative has a repeated root, none when no real roots.

The move. A cubic with a positive leading coefficient always has its local maximum on the left and its local minimum on the right. (Negative leading coefficient reverses the order.) Use that as a sanity check before reaching for the second derivative.