In Exercises 37-40 , locate the absolute extrema of the...

In Exercises $37-40$ , locate the absolute extrema of the function (if any exist) over each interval.
$$
\begin{array}{l}{f(x)=x^{2}-2 x} \\ {\text { (a) }[-1,2] \quad \text { (b) }(1,3]} \\ {\text { (c) }(0,2) \quad \text { (d) }[1,4)}\end{array}
$$

Key Concepts

Absolute Extrema

Absolute extrema refer to the overall highest or lowest values that a function attains on a given interval. Finding these values involves comparing the function’s output at candidate points such as critical points and endpoints (if they are part of the interval).

Critical Points

Critical points are values in the domain of the function where the derivative is zero or does not exist. These points are important because they are potential locations where the function might attain local or absolute extrema.

Closed and Open Intervals

The type of interval considered in a problem is crucial because the existence of extrema can depend on whether the interval is closed (includes its endpoints) or open (excludes its endpoints). The Extreme Value Theorem guarantees that a continuous function on a closed interval has both an absolute maximum and an absolute minimum, while this may not be the case for open intervals.

Differentiation

Differentiation is used to determine the derivative of the function, which helps locate critical points. By setting the derivative equal to zero or finding where it fails to exist, one identifies candidate points for potential extrema.

Comparing Function Values

After identifying the critical points and considering any endpoints included in the interval, it is necessary to evaluate the function at these points. The absolute extremum is determined by comparing these values, and the highest and lowest values represent the absolute maximum and minimum respectively.

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