Calculus How To

Stationary Point

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What is a Stationary Point?

A stationary point is the point at which the derivative is zero; where

f'(x0)= 0.

Stationary points include minimums, maximums, and inflection points; but not all inflection points are stationary points.

stationary point

The stationary points are the red circles. In this graph, they are all relative maxima or relative minima. The blue squares are inflection points.

Stationary points are called that because they are the point at which the function is, for a moment, stationary: neither decreasing or increasing.

Using Stationary Points for Curve Sketching

Stationary points can help you to graph curves that would otherwise be difficult to solve. They include most of the interesting points on the curve, and if you graph them, and connect the dots, you have a fairly good general curve of your function.

You will want to know, before you begin a graph, whether each point is a maximum, a minimum, or simply an inflection point. One way to do this is by looking at the second derivative.

If the second derivative is less than 0, the stationary point is a maximal extremum, and the graph is concave down right at that point.

  • If the second derivative is more than 0, the stationary point is a minimal extremum, and the graph is concave up at that point.
  • If the second derivative is equal to zero, you’ll have to use other methods to determine what kind of stationary point it is. You might look at the sign change around that point, for instance.


Patrikalakis, Maekawa, & Cho. Classification of stationary points. 7.3.1 in Shape Interrogation for Computer Aided Design and Manufacturing (Hyperbook Edition) Retrieved from on Feb 19, 2019

University of East Anglia Learning Enhancement Team. Steps Into Calculus: Finding Stationary Points. Retrieved from on Feb 20, 2019.

Stephanie Glen. "Stationary Point" From Calculus for the rest of us!

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