02 Stationary points and the first-derivative test
A stationary point is a point on the curve where f′(x) = 0; the tangent there is horizontal (parallel to the x-axis).
- Local maximum: f′ changes from positive to negative as x increases through the point (the curve rises then falls).
- Local minimum: f′ changes from negative to positive.
- Point of inflection (stationary): f′ does not change sign — it keeps the same sign on both sides, e.g. f(x) = x³ at x = 0.
Worked example
f(x) = x³ − 6x² + 9x + 2, stationary at x = 1 and x = 3.
- At x = 1 the sign goes + → −, so maximum at (1, f(1)) = (1, 6).
- At x = 3 the sign goes − → +, so minimum at (3, 2).
Note that a maximum value is local: the local maximum 6 is smaller than the value at x = 5, which is 27.