A relative minimum is a point on a function’s graph where the output value is lower than all nearby points. Think of it as the bottom of a valley: the function decreases as it approaches this point, then increases as it moves away. The formal term “relative” (or “local”) means you’re only comparing the point to its immediate neighbors, not to every point on the entire graph.
The Formal Definition
A point (x, f(x)) is a relative minimum if there exists some open interval around x where f(x) is less than or equal to f(z) for every z in that interval. In plain terms, you can draw a small window around the point, and nothing inside that window sits lower on the graph. The value doesn’t need to be the lowest the function ever reaches. It only needs to be the lowest in its neighborhood.
You’ll see “relative minimum” and “local minimum” used interchangeably in textbooks and online resources. They mean exactly the same thing.
Relative Minimum vs. Absolute Minimum
An absolute minimum (also called a global minimum) is the single lowest value a function produces across its entire domain. A relative minimum just needs to be the lowest point in some small surrounding region. A function can have several relative minima, but it can have at most one absolute minimum value. One of the relative minima might also be the absolute minimum, but that’s not guaranteed.
Picture a mountain range viewed from the side. Each valley floor is a relative minimum. The deepest valley of all is the absolute minimum. The other valleys are still relative minima even though they sit higher than the deepest one.
What It Looks Like on a Graph
On a graph, a relative minimum appears as a valley or trough. The curve slopes downward as it approaches the point, bottoms out, then slopes upward as it leaves. If you traced the curve with your finger from left to right, your finger would move down, reach the lowest nearby spot, and then start moving back up.
This visual pattern corresponds to a specific behavior in the slope of the function. To the left of the valley, the slope is negative (the function is falling). At the bottom of the valley, the slope is zero or undefined. To the right, the slope turns positive (the function is rising). That shift from negative slope to positive slope is the signature of a relative minimum.
Critical Points: Where Relative Minima Can Occur
Relative minima can only occur at what mathematicians call critical points. A critical point is an interior point in the function’s domain where either the derivative equals zero or the derivative doesn’t exist. “Interior” is key here: endpoints of a closed interval are not critical points, so by this convention, relative extrema cannot occur at endpoints.
Not every critical point produces a relative minimum. Some produce relative maxima (hilltops), and some produce neither, like the flat spot on a curve that briefly levels out before continuing in the same direction. To determine whether a critical point is actually a relative minimum, you need one of two tests.
The First Derivative Test
The first derivative test checks the sign of the derivative on either side of a critical point. Here’s how it works:
- Find the critical point. Set the derivative equal to zero (or identify where it’s undefined) and solve for x.
- Check the sign of the derivative to the left. Pick a value slightly less than your critical point and plug it into the derivative. If the result is negative, the function is decreasing there.
- Check the sign of the derivative to the right. Pick a value slightly greater than the critical point and plug it in. If the result is positive, the function is increasing there.
If the derivative changes from negative to positive at your critical point, you have a relative minimum. The function was going down, and now it’s going up, so you’re at the bottom of a valley. If the sign change goes the other direction (positive to negative), that’s a relative maximum instead. If there’s no sign change at all, the critical point is neither.
The Second Derivative Test
The second derivative test offers a shortcut when the second derivative is easy to compute. Instead of checking signs on both sides, you evaluate the second derivative at the critical point itself.
If the second derivative at that point is positive, the function is concave up there, meaning it curves like the inside of a bowl. A bowl shape with a flat tangent line at the bottom is a valley, so the point is a relative minimum. If the second derivative is negative, the function is concave down (like an upside-down bowl), and the point is a relative maximum instead.
There’s one limitation: if the second derivative equals zero, the test is inconclusive. You’d need to fall back on the first derivative test or analyze higher-order derivatives to classify the point.
A Worked Example
Consider the function f(x) = x² − 4x + 5. The derivative is f′(x) = 2x − 4. Setting this equal to zero gives x = 2, so x = 2 is a critical point.
Using the first derivative test: pick x = 1 (to the left), and f′(1) = 2(1) − 4 = −2, which is negative. Pick x = 3 (to the right), and f′(3) = 2(3) − 4 = 2, which is positive. The derivative changes from negative to positive, confirming a relative minimum at x = 2. Plugging back in, f(2) = 4 − 8 + 5 = 1, so the relative minimum is the point (2, 1).
You can verify with the second derivative test: f″(x) = 2, which is positive everywhere. Since f″(2) > 0, the function is concave up at x = 2, confirming the relative minimum. In this particular case, because it’s a simple parabola opening upward, this relative minimum is also the absolute minimum of the function.
When the Derivative Doesn’t Exist
Relative minima don’t always occur at smooth, rounded valley bottoms. They can also appear at sharp corners or cusps where the derivative is undefined. The classic example is f(x) = |x|, which has a sharp V-shape at x = 0. There’s no single tangent line at the tip, so the derivative doesn’t exist there, but x = 0 is still a critical point and a relative minimum. The function decreases on the left side and increases on the right side, satisfying the same sign-change condition.
This is why the definition of a critical point includes both cases: derivative equals zero and derivative undefined. If it only covered the zero case, sharp-cornered minima would slip through.