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

In Exercises $29-58,$ find the minimum and maximum value of the function on the given interval by comparing values at the critical points and endpoints.
$$y=x^{3}-24 \ln x, \quad\left[\frac{1}{2}, 3\right]$$

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Key Concepts

Endpoint Evaluation

When determining the absolute maximum and minimum values of a function on a closed interval, it is crucial to evaluate the function at the endpoints of the interval along with any critical points. This is necessary because the extreme values of the function might occur at the boundaries of the interval.

Comparison Method for Absolute Extrema

This method involves calculating the function's values at all relevant points—critical points and endpoints—and then directly comparing these values to determine which are the highest and lowest. This systematic approach ensures that the absolute maximum and minimum values are accurately identified.

Extreme Value Theorem

This theorem states that a continuous function on a closed and bounded interval will always attain both an absolute maximum and an absolute minimum. It forms the fundamental basis for evaluating the extreme values of a function over a specific interval.

Critical Points

Critical points are the points in the domain of a function where the derivative is zero or undefined. They are important in optimization problems because these are the potential candidates for local extrema, which may also be absolute extrema when analyzed over a closed interval.

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