Final answer:
Critical points, inflection points, and stationary points are determined through the analysis of the first and second derivatives of a function. Critical points are where the first derivative is zero or undefined, possibly indicating local extrema. Stationary points are where the first derivative is zero; they can be local maxima, minima, or saddle points, while inflection points occur where the concavity changes.
Step-by-step explanation:
Understanding Critical Points, Inflection Points, and Stationary Points
To determine points of interest such as critical points, inflection points, and stationary points, on a graph of a function, it is essential to analyze the derivative of the function. Critical points occur where the first derivative is zero or undefined. These points are where the graph may change its direction of increase or decrease and can indicate potential local maxima or minima.
Inflection Points and How to Find Them
Inflection points are points on the graph where the concavity changes, which means the second derivative changes sign. To find the inflection points, set the second derivative equal to zero and solve for the variable.
Identifying Stationary Points
Stationary points are points on the graph where the first derivative is zero. These can be local maxima, minima, or saddle points depending on the behavior of the first and second derivatives around these points.
Saddle points are types of stationary points where the function does not achieve a local maximum or minimum, instead, the slopes of the tangent lines on either side of the point are of opposing signs.
To thoroughly explore these concepts and how they apply to graphs, one would typically use calculus techniques including differentiation and, possibly, the second derivative test for concavity and inflection.