Question
on a graph what is an extreme value
Answers
GPT-4o mini
In the context of a graph, an extreme value refers to a point on a curve that represents a local maximum or minimum, where the function reaches its highest or lowest value within a given neighborhood.
1. **Local Maximum**: A point \( x = c \) is a local maximum if \( f(c) \) is greater than the values of \( f(x) \) for all \( x \) in some interval around \( c \). In other words, it is the highest point in the vicinity.
2. **Local Minimum**: A point \( x = c \) is a local minimum if \( f(c) \) is less than the values of \( f(x) \) for all \( x \) in some interval around \( c \). This means it is the lowest point in the vicinity.
3. **Global Maximum and Minimum**: In addition to local extreme values, a function can also have global (or absolute) maximum and minimum values, which are the highest and lowest points overall in the entire domain of the function.
Extreme values are important in various fields, including calculus, optimization, and various applications in economics, engineering, and the sciences, as they can provide insights into the behavior and properties of functions.
1. **Local Maximum**: A point \( x = c \) is a local maximum if \( f(c) \) is greater than the values of \( f(x) \) for all \( x \) in some interval around \( c \). In other words, it is the highest point in the vicinity.
2. **Local Minimum**: A point \( x = c \) is a local minimum if \( f(c) \) is less than the values of \( f(x) \) for all \( x \) in some interval around \( c \). This means it is the lowest point in the vicinity.
3. **Global Maximum and Minimum**: In addition to local extreme values, a function can also have global (or absolute) maximum and minimum values, which are the highest and lowest points overall in the entire domain of the function.
Extreme values are important in various fields, including calculus, optimization, and various applications in economics, engineering, and the sciences, as they can provide insights into the behavior and properties of functions.