Find the absolute maximum and minimum values of each...

Find the absolute maximum and minimum values of each function, if they exist, over the indicated interval. Also indicate the $x$ -value at which each extremum occurs. When no interval is specified, is specified, use the real line, $(-\infty, \infty)$. $$f(x)=\frac{1}{3} x^{3}-x+\frac{2}{3}$$

Key Concepts

Absolute Extrema

This concept refers to the highest (maximum) and lowest (minimum) values that a function attains on a specified domain. Identifying absolute extrema requires analyzing the entire domain of the function, including any endpoints if the domain is closed and bounded, or evaluating the behavior of the function at infinity when the domain is unbounded.

Critical Points

Critical points of a function are points in its domain where the derivative is zero or does not exist. These points are important because local maximums or minimums often occur at critical points, making them key candidates when searching for absolute extrema.

Derivative

The derivative of a function represents its instantaneous rate of change. It is used to identify critical points by setting it equal to zero. In optimization problems, calculating the derivative helps in determining where the function may change direction, thus pinpointing potential extreme values.

Extreme Value Theorem

The Extreme Value Theorem guarantees that a continuous function on a closed and bounded interval will have both an absolute maximum and an absolute minimum. This theorem underpins many optimization problems, though its assurance does not extend to open or unbounded intervals.

Behavior on Unbounded Domains

When a function is defined on an unbounded domain, such as the entire real line, it is essential to evaluate its limits as the variable approaches infinity or negative infinity. This analysis helps determine whether the function has absolute extrema or if it diverges, as the extreme value theorem does not apply in such contexts.

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