Stationary Point — Definition, Formula & Examples
A stationary point is a point on a curve where the derivative equals zero, meaning the tangent line is horizontal. At a stationary point, the function is momentarily neither increasing nor decreasing.
A point in the domain of a differentiable function is a stationary point if . Stationary points are a subset of critical points, which also include points where the derivative is undefined.
Key Formula
Where:
- = The first derivative of f evaluated at x = c
- = The x-coordinate of the stationary point
Worked Example
Problem: Find the stationary points of f(x) = x³ − 6x² + 9x + 2.
Differentiate: Compute the first derivative of f.
Set derivative to zero: Solve f'(x) = 0 by factoring.
Find coordinates: The solutions are x = 1 and x = 3. Evaluate f at each: f(1) = 1 − 6 + 9 + 2 = 6 and f(3) = 27 − 54 + 27 + 2 = 2.
Answer: The function has stationary points at (1, 6) and (3, 2).
Why It Matters
Stationary points identify where a function has local maxima, local minima, or inflection points with horizontal tangents. In physics, setting the velocity function equal to zero finds the moments when an object changes direction. Curve sketching problems on the AP Calculus exam rely heavily on locating stationary points.
Common Mistakes
Mistake: Treating "stationary point" and "critical point" as identical.
Correction: Every stationary point is a critical point, but not every critical point is stationary. A critical point can also occur where the derivative is undefined (such as a cusp), whereas a stationary point specifically requires f'(c) = 0.