## Extreme Value Theoremf is cont. in the closed interval [a,b] then…

### If a function f is continuous in the closed interval [a, b], then the Extreme Value Theorem guarantees the existence of both a maximum and a minimum value of the function in that interval

If a function f is continuous in the closed interval [a, b], then the Extreme Value Theorem guarantees the existence of both a maximum and a minimum value of the function in that interval.

According to the theorem, the function f must be defined on the entire closed interval [a, b], meaning it must have values for every x between a and b. Additionally, f must be continuous, meaning that it has no “breaks” or jumps in its graph within the interval.

The Extreme Value Theorem states that the function f will have both a maximum and a minimum value within the interval [a, b]. This means that there will be a point c in the interval where f(c) is the largest value attained by the function, and there will also be a point d in the interval where f(d) is the smallest value attained.

It’s important to note that the theorem guarantees the existence of these maximum and minimum values, but it does not specify their exact locations or give a formula to find them. In order to find the precise maximum or minimum value, further analysis such as finding critical points or using derivative techniques may be necessary. However, the Extreme Value Theorem assures us that such values do exist within the given interval for a continuous function.

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