In Exercises 17-36, locate the absolute extrema of the function on the closed interval. y=tan((π

step 1 In this question, we will need to find the absolute extremal of the function. We call that the a

In Exercises 17-36, locate the absolute extrema of the...

Key Concepts

Extreme Value Theorem

The Extreme Value Theorem states that if a function is continuous on a closed interval, then it must have both an absolute maximum and an absolute minimum on that interval. This concept is fundamental when searching for extrema, as it guarantees that the search for the highest and lowest values on the given interval is well-founded.

Critical Points

Critical points are values in the domain of a function where the derivative is zero or undefined. In the context of finding extrema on a closed interval, these points are candidates for local minima and maxima. Evaluating the function at these points, as well as at the interval’s endpoints, is essential in determining the absolute extrema.

Differentiation to Identify Candidates

Finding the derivative of the function helps identify critical points by setting the derivative equal to zero or checking where the derivative does not exist. This process is a key step in analyzing the function’s behavior and systematically locating where potential extrema might occur.

Endpoint Evaluation

When determining absolute extrema on a closed interval, it is important to evaluate the function at the interval's endpoints. Since local extrema can occur at these boundaries, comparing the function's values at the endpoints with those at the critical points helps ensure that the absolute maximum and minimum are correctly identified.

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