What is the first derivative test used for?

The first derivative test is a method used to identify local maxima and minima of a function. When finding critical points—where the first derivative is either zero or undefined—you can analyze the behavior of the derivative before and after these points. If the derivative changes from positive to negative at a point, that point is a local maximum. Conversely, if the derivative changes from negative to positive, it indicates a local minimum.

This test is particularly useful because it provides insight into how the function behaves in the neighborhood of the critical points, allowing you to classify those points effectively. Understanding local maxima and minima is vital in optimization problems, where determining the highest or lowest values is often necessary.

Other options relate to different concepts. For instance, determining the slope of a function pertains to the value of the derivative itself rather than its implications for maxima and minima. Finding points of intersection involves solving equations rather than analyzing the critical points. Analyzing concavity is a function of the second derivative, which focuses on the curvature of the graph rather than the identification of local extrema.