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Calculus, a vast branch of mathematics, primarily deals with the study of change and motion. Through its two major subdivisions, differential calculus and integral calculus, it allows us to compute rates of change (derivatives) and the total accumulation of quantities (integrals). For instance, in real-world problems, calculus helps in understanding the variations in system dynamics and can aid in finding solutions by analyzing the behavior of functions.

In our exercise, calculus lays the groundwork for locating absolute extrema, which are, essentially, the highest or lowest values a function can attain on a given interval. This calculus-based search for extrema involves finding derivatives to determine critical points and analyzing the behavior of the function at the endpoints of intervals.

The Extreme Value Theorem states that a continuous function on a closed interval has both a maximum and a minimum value. In other words, if you have a graph that represents a continuous function without any breaks or holes, and you look at a segment of this graph that starts and ends within the domain of the function, there will be both a highest and lowest point on this segment.

Looking at our exercise, the theorem can be applied to part (a) where the interval is closed, i.e., \[0,2\]. Due to the function being both continuous and the interval being closed, we are guaranteed to have absolute maximum and minimum values, thus requiring an assessment of both endpoints of the interval.

Closed intervals are intervals that include their endpoints. In notation, a closed interval from 'a' to 'b' is written as \[a,b\]. Such intervals are of special interest in calculus because, as mentioned, the Extreme Value Theorem assures us that continuous functions will exhibit an absolute maximum and minimum within these boundaries.

Applying this concept to part (a) of our exercise with interval \[0,2\], we observed that both endpoints are considered to find the extrema, heightening the importance of endpoint evaluation whenever coping with closed intervals.

Critical points in calculus are points on a graph where the derivative of the function is either zero or undefined. These points are crucial in determining where a function could have possible maxima or minima. To find critical points, one typically takes the derivative of the function and sets it equal to zero or looks for where the derivative may not exist.

In the context of our linear function, \(f(x) = 2x - 3\), it does not have any critical points since the derivative, \(f'(x) = 2\), is never zero nor undefined. However, for more complex functions, locating and analyzing critical points are indispensable steps in analyzing the function's behavior and thereby determining its extrema.