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Critical Points

Definition

Critical points are the x-values in a function where the derivative is either zero or undefined. They represent potential locations of maximum, minimum, or inflection points.

Analogy

Think of critical points as crossroads on a map. At these crossroads, you have to make important decisions about which direction to take. Similarly, critical points in a function indicate significant changes in its behavior.

Related terms

Local Maximum/Minimum: A local maximum (or minimum) occurs at a critical point where the function reaches its highest (or lowest) value within a specific interval but may not be the overall maximum (or minimum).

Inflection Point: An inflection point is a critical point where the concavity of the graph changes from concave up to concave down or vice versa.

First Derivative Test: The first derivative test is used to determine whether a critical point corresponds to a local maximum, local minimum, or neither by analyzing the sign changes of the derivative around that point.