In Exercises 51-58 , find the critical points and domain...

In Exercises $51-58$ , find the critical points and domain endpoints for
each function. Then find the value of the function at each of these
points and identify extreme values (absolute and local).
$$
y=x^{2 / 3}(x+2)
$$

Key Concepts

Derivative and Its Use in Finding Extrema

The derivative of a function provides information about the rate of change of the function. By finding where the derivative equals zero or is undefined, one can locate the critical points, which are potential candidates for local extrema. Additionally, derivative tests help determine whether these points correspond to local maxima, minima, or neither.

Domain of a Function

The domain of a function is the complete set of all possible input values (typically x-values) for which the function is defined. Determining the domain involves identifying restrictions such as division by zero, square roots of negative numbers, or any other limitations imposed by the function’s formula.

Critical Points

Critical points occur at values within the domain of the function where the derivative is zero or does not exist. These points are significant because they often correspond to local maximums, minimums, or points of inflection and are a key focus when analyzing the behavior of a function.

Endpoints of the Domain

Endpoints are the boundary values of the domain, especially in cases where the domain is restricted to a closed interval. Since extreme values can occur at these endpoints, they must be evaluated along with the critical points when looking for absolute maximum or minimum values.

Extreme Values and the Extreme Value Theorem

Extreme values refer to the absolute highest and lowest values that a function attains on a given interval. 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, which will occur either at critical points or at the endpoints of the domain.

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