A local minimum on a graph is a point where the y-value is lower than all nearby points. Visually, it looks like the bottom of a valley: the graph decreases as it approaches the point, then increases as it moves away. A function can have one local minimum, several, or none at all.
The formal way to say this: a function f(x) has a local minimum at x = c if f(x) ≥ f(c) for every x in some open interval around c. The key phrase is “some open interval.” You don’t need the point to be the lowest value on the entire graph. You just need a small neighborhood where nothing dips below it.
How to Spot One on a Graph
Local minima appear as valleys. The graph is heading downward (decreasing), hits a low point, then turns around and heads upward (increasing). That turning point is the local minimum. If you placed a marble on the curve at that spot, it would roll uphill in either direction.
A function can have multiple valleys at different heights. A graph might dip to y = 1 in one spot and dip to y = 3 in another. Both are local minima because each one is the lowest point in its own neighborhood, even though y = 3 is clearly higher than y = 1. To find all the local minima on a graph, look for every point where the curve switches from going down to going up.
Local Minimum vs. Global Minimum
A global (or absolute) minimum is the single lowest point on the entire graph. A local minimum only needs to be the lowest point in its immediate area. Every global minimum is automatically a local minimum, but not every local minimum is the global one.
Think of hiking through a mountain range. You might descend into several valleys along the way. Each valley floor is a local minimum. But only the deepest valley in the entire range is the global minimum. If the terrain has just one valley and no other dips, then that local minimum and the global minimum are the same point.
There is one special case worth knowing: if a function forms a smooth, bowl-shaped curve (mathematically called a convex function), then any local minimum is guaranteed to also be the global minimum. There’s only one valley, and it’s the deepest point everywhere.
Finding Local Minima With Calculus
If you’re working with a function’s equation rather than just looking at a picture, calculus gives you two reliable tools for confirming a local minimum.
The First Derivative Test
The first derivative, f'(x), tells you whether the function is increasing or decreasing. At a local minimum, the derivative changes from negative (the function is falling) to positive (the function is rising). This sign change is the mathematical fingerprint of a valley. You find the points where f'(x) = 0 or where f'(x) doesn’t exist, then check whether the derivative switches from negative to positive at that point.
The Second Derivative Test
The second derivative, f”(x), tells you about the curve’s shape. If f'(c) = 0 and f”(c) > 0, there is a local minimum at x = c. The logic is intuitive: f”(c) > 0 means the graph is curving upward at that point, forming a bowl shape. The critical point sits at the bottom of that bowl.
If f”(c) = 0 or doesn’t exist, this test gives no answer, and you need to fall back on the first derivative test or inspect the graph directly.
Local Minima at Sharp Corners
Local minima don’t always occur at smooth, rounded valley bottoms. They can also appear at sharp points where the derivative is undefined. The classic example is f(x) = |x|, the absolute value function. Its graph forms a V-shape with the lowest point at x = 0. The function decreases on the left side and increases on the right side, making x = 0 a local minimum, but the graph has a sharp corner there rather than a smooth curve. The derivative doesn’t exist at that point because the slope changes abruptly from negative to positive with no gradual transition.
This is why the formal definition of a critical point includes places where the derivative equals zero or is undefined. Both types of points are candidates for local minima. When you’re scanning a graph, look for valleys with smooth bottoms and valleys with pointed bottoms alike.
Practical Examples
Consider the function f(x) = x². Its graph is a parabola opening upward, with a single valley at x = 0. That point is both a local minimum and the global minimum. The first derivative is 2x, which is negative for x < 0 and positive for x > 0, confirming the sign change. The second derivative is 2, which is positive everywhere, confirming the upward bowl shape.
Now consider f(x) = x³ – 3x. This function has a local minimum at x = 1, where the graph dips to y = -2, and a local maximum at x = -1, where the graph peaks at y = 2. The curve snakes up and down, creating one hill and one valley. The valley at (1, -2) is the local minimum because the function decreases into that point and increases away from it. But this function extends to negative infinity as x goes left, so there is no global minimum at all.
The important takeaway: a local minimum describes the shape of the graph in a specific neighborhood. It tells you where a valley is, regardless of what the function does far away from that point.