A stationary point is a point on a curve where the slope equals zero. In calculus terms, it’s any point where the first derivative of a function is zero, meaning the tangent line to the curve is perfectly horizontal at that location. The function momentarily “stops” increasing or decreasing, even if only for an instant.
Why the Slope Equals Zero
The derivative of a function tells you how steeply the curve is rising or falling at any given point. A positive derivative means the function is increasing; a negative derivative means it’s decreasing. At a stationary point, the derivative is exactly zero, so the function is neither climbing nor dropping. Visually, if you drew a tangent line touching the curve at that point, it would be completely flat.
To find stationary points, you take the derivative of a function and set it equal to zero, then solve for the input values. Those values are your stationary points. For example, the function f(x) = x² has a derivative of 2x, which equals zero when x = 0. So x = 0 is a stationary point, and the curve bottoms out there.
The Three Types of Stationary Points
Not all stationary points look the same on a graph. There are three distinct types, and each tells you something different about the curve’s behavior.
Local maximum: The function reaches a peak. It was increasing on the left side of this point and decreasing on the right. Think of the top of a hill. The slope goes from positive to negative.
Local minimum: The function reaches a valley. It was decreasing before this point and increasing after. The slope goes from negative to positive.
Stationary point of inflection: The slope hits zero, but the function doesn’t actually change direction. It was increasing before, flattens briefly, and then continues increasing (or the same pattern with decreasing). The curve changes its concavity, switching from curving upward to curving downward or vice versa, but keeps heading the same way. The function f(x) = x³ at x = 0 is a classic example.
The First Derivative Test
Once you’ve found a stationary point by solving for where the derivative equals zero, you need to figure out which type it is. The first derivative test does this by checking the sign of the derivative on either side of the point.
- Positive then negative: The function rises into the point and falls away from it. That’s a local maximum.
- Negative then positive: The function falls into the point and rises away. That’s a local minimum.
- No sign change: The derivative is positive on both sides (or negative on both sides). The function didn’t reverse direction, so this is a stationary point of inflection.
In practice, you pick a value slightly to the left and slightly to the right of the stationary point, plug each into the derivative, and check whether the result is positive or negative. The pattern of signs tells you everything.
The Second Derivative Test
The second derivative test is often faster. Instead of checking the slope on both sides, you take the second derivative of the function (the derivative of the derivative) and evaluate it at the stationary point.
If the second derivative is negative at that point, the curve is concave down (shaped like an upside-down bowl), so you’re at a local maximum. If the second derivative is positive, the curve is concave up (shaped like a bowl), and you’re at a local minimum.
If the second derivative is also zero, the test is inconclusive. This can happen at a stationary point of inflection, but it can also occur at some maxima and minima. When the second derivative test fails, you need to fall back on the first derivative test and check the sign changes directly.
Stationary Points vs. Critical Points
These two terms overlap but aren’t identical. A stationary point is specifically where the derivative equals zero. A critical point is a broader category that includes stationary points plus any point where the derivative doesn’t exist at all, like a sharp corner or cusp on a graph. A function can have a local maximum at a sharp corner even though there’s no horizontal tangent there.
Some textbooks (particularly in the U.S.) use “critical point” and “stationary point” interchangeably, but strictly speaking, every stationary point is a critical point, while not every critical point is a stationary point. When you’re searching for the absolute highest or lowest value of a function, you need to check both: the places where the derivative is zero and the places where the derivative doesn’t exist.
Stationary Points With Multiple Variables
For functions with more than one input variable, the idea extends naturally. A stationary point occurs where all the partial derivatives are zero simultaneously, meaning the gradient vector (which collects all the partial derivatives into a single vector pointing in the direction of steepest ascent) equals zero. At that point, the function isn’t increasing in any direction.
Classification gets more complex with multiple variables because a new type appears: the saddle point. A saddle point is stationary (zero gradient) but is a maximum in one direction and a minimum in another, like the center of a horse saddle or a mountain pass. To classify these, you use a tool called the Hessian matrix, which organizes all the second-order partial derivatives. If the Hessian’s properties indicate mixed positive and negative curvature, you have a saddle point rather than a true maximum or minimum.
Where Stationary Points Show Up
Finding stationary points is the core technique behind optimization, which is the process of finding the best possible value of something. Any time you want to maximize profit, minimize cost, find the most efficient shape, or determine the lowest energy state of a physical system, you’re looking for stationary points.
In physics, objects tend to settle into positions where potential energy is at a minimum, which is a stationary point. In engineering, designing a container that uses the least material for a given volume is a minimization problem solved by finding where the derivative of the material function equals zero. In economics, firms minimizing production costs or maximizing output are solving for stationary points of cost or revenue functions. Machine learning algorithms that train neural networks are essentially searching for minima of error functions across thousands or millions of variables, all guided by the same principle: find where the gradient is zero.