What does a critical point signify in an optimization problem?

In the context of optimization problems, a critical point is defined as a location in the function where the derivative is either zero or undefined. This is significant because such points are where the function may reach a local maximum or minimum. When optimizing a function, identifying these critical points is essential because they are potential candidates for the best solutions—either the highest or lowest values of the function.

At a critical point where the derivative is zero, it indicates that there is no slope in that vicinity, suggesting that the function may be leveling out, which is characteristic of maxima, minima, or, in some cases, points of inflection. If a derivative is undefined, it generally indicates that the function may have a sharp point or a vertical tangent, which can also represent a local extremum. Therefore, recognizing critical points aids in determining where to evaluate a function to find optimal values.