What are two conditions that must be met before the Extreme Value Theorem may be applied?
no ideas? Try reading the theorem.
To determine the conditions that must be met before the application of the Extreme Value Theorem, we need to have a clear understanding of what the theorem states.
The Extreme Value Theorem states that if a function f(x) is continuous on a closed interval [a, b], then there exist two points within the interval, denoted as c and d, where f(c) is the maximum value and f(d) is the minimum value of the function.
Now, let's discuss the two conditions that must be met for the Extreme Value Theorem to be applicable:
1. Continuity: The function f(x) must be continuous on the closed interval [a, b]. Continuity means that there are no jumps, breaks, or any other discontinuities in the function over the interval. In simpler terms, the graph of the function should not have any gaps, holes, or vertical asymptotes within the interval [a, b].
2. Closed Interval: The interval [a, b] must be closed, meaning that it includes its endpoints. Both the starting point, a, and the ending point, b, are part of the interval. In other words, the interval should not be open (where one or both endpoints are excluded) or half-open (where only one endpoint is included).
To apply the Extreme Value Theorem, these two conditions, continuity of the function and the closed interval, must be satisfied. If both conditions are met, we can conclude that there exist points c and d within the interval [a, b] where f(c) is the maximum value and f(d) is the minimum value.