In Exercises 29-58, find the minimum and maximum value of the function on the given interval by

step 1 Question 55 we need to find the maximum and minimum value of this function in this interval usin

In Exercises 29-58, find the minimum and maximum value of...

Key Concepts

Critical Points

Critical points are values in the domain of a function where its derivative is zero or does not exist. These points are important because they indicate where the function’s rate of change switches behavior, and they are the possible locations for local extrema, which need to be considered when determining the overall maximum and minimum values on an interval.

Extreme Value Theorem

The Extreme Value Theorem states that a continuous function on a closed interval must attain both a maximum and a minimum value. This theorem underpins the method of evaluating the function at critical points and at the endpoints of the interval to find these global extrema.

Derivative Analysis

Differentiation is used to compute the derivative of the function, which helps identify the critical points by solving for where the derivative equals zero or is undefined. This is the first necessary step in the process of optimizing the function.

Endpoint Evaluation

Evaluating the function at the endpoints of the closed interval is essential because, according to the Extreme Value Theorem, the global maximum or minimum may occur at these boundary points rather than at the critical points found within the interior of the interval.

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