Which of the following most accurately identifies the types of extrema in the graph?
(1 point)
Responses
There is one relative maximum and one absolute minimum.
There is one relative maximum and one absolute minimum.
There is one relative maximum and one relative minimum.
There is one relative maximum and one relative minimum.
There is one absolute maximum and one relative minimum.
There is one absolute maximum and one relative minimum.
There is one relative maximum, one relative minimum, and one absolute minimum.
There is one relative maximum and one relative minimum.
To identify the types of extrema in the graph, we need to understand their definitions.
1) A relative maximum or minimum is a point where the function reaches its highest or lowest value in a certain interval, but it may not be the highest or lowest value overall.
2) An absolute maximum or minimum is a point where the function reaches its highest or lowest value on the entire domain.
Based on the given options, the most accurate identification would be:
"There is one relative maximum, one relative minimum, and one absolute minimum."
This means that there is a point on the graph that is the highest in its interval (relative maximum), another point that is the lowest in its interval (relative minimum), and a third point that is the lowest on the entire domain (absolute minimum).
To identify the types of extrema in a graph, we need to understand the concepts of relative extrema and absolute extrema.
A relative extremum occurs when the graph of a function reaches a peak or a valley locally. In other words, it is the highest or lowest point in a particular region of the graph. A relative maximum is the highest point in a region, while a relative minimum is the lowest point in a region.
An absolute extremum, on the other hand, refers to the highest or lowest point of the entire graph. It is the global maximum or minimum.
To determine the extrema in a graph, we can find the critical points by taking the derivative of the function and setting it equal to zero. Critical points are the potential locations of extrema. We then analyze the behavior of the function on both sides of each critical point to determine if it is a relative maximum or minimum.
Now, let's analyze the answer choices:
1. There is one relative maximum and one absolute minimum.
2. There is one relative maximum and one relative minimum.
3. There is one absolute maximum and one relative minimum.
4. There is one relative maximum, one relative minimum, and one absolute minimum.
From the answer choices provided, it seems that only option 4 includes all three types of extrema (relative maximum, relative minimum, and absolute minimum). Therefore, the most accurate option for identifying the types of extrema in the graph is option 4: "There is one relative maximum, one relative minimum, and one absolute minimum."