Extreme Value Theorem: A key theorem in calculus regarding continuous functions.
Purpose: Determines conditions under which a function is guaranteed to have a maximum and minimum.
Limitation: Does not provide the method to calculate these extrema, only assures their existence.
Definitions
Maximum of a Function
A function has a maximum at a point if, at that point, the function's value, say ( f(c) ), is greater than or equal to its value at any other point in the domain.
Formal Definition: There exists a ( c ) in the domain ( I ) such that for all ( x ) in ( I ), ( f(x) \leq f(c) ).
Note: The maximum is the value ( f(c) ), occurring at input ( c ).
Minimum of a Function: Similar definition structure to the maximum.
Understanding Extrema Through Examples
Example 1: Function with a maximum inside an interval.
Example 2: Function with a maximum at an endpoint.
Example 3: Function without a maximum due to an open interval.
Example 4: Function without a maximum, with an unbounded domain and vertical asymptote.
Observations from Examples
Closed and Bounded Intervals: Functions on closed intervals with both endpoints included tend to have maxima and minima.
Open Intervals and Discontinuities: Functions on open intervals or with discontinuities may lack maxima or minima.
Extreme Value Theorem
Statement: If ( f ) is continuous on a closed and bounded interval ([a, b]), then ( f ) must have both a maximum and a minimum.
Proof: Although not covered, the theorem's proof is noted as complex and typically covered in advanced analysis courses.
Challenge
Construct a Function:
That is continuous on an interval missing one endpoint (e.g., excluding 0, including 1).
Does not have a maximum or minimum.
Hint: Visualize a graph, then derive an equation.
Conclusion
The theorem offers a foundational understanding that helps predict the behavior of continuous functions on closed intervals.
Practical implications in various fields of mathematics and applied sciences.