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

a. Find the absolute maximum and minimum values of each function on the given interval.
b. Graph the function, identify the points on the graph where the absolute extrema occur, and include their coordinates.
$g(x)=e^{-x^2}, \quad-2 \leq x \leq 1$

Key Concepts

Graphical Analysis

Graphing the function aids in visualizing its behavior, confirming the results obtained analytically, and understanding where key features such as the absolute maximum and minimum occur. It also helps in illustrating the symmetry or specific shape characteristics of the function.

Evaluation at Endpoints

In problems involving functions on closed intervals, evaluation at the endpoints is crucial, as the absolute extrema might occur there even if the derivative does not vanish. Comparing the values at the endpoints with those determined from the critical points ensures finding the global extrema.

Differentiation

Computing the derivative of the function is a key step in determining where the function's slope is zero or undefined. This process allows one to identify critical points, which are necessary for investigating the local behavior of the function.

Critical Points Analysis

Finding the critical points by setting the derivative equal to zero is essential in locating potential local extrema within the interval. These critical points are candidates to produce absolute maximum or minimum values when compared with the function's values at the interval's endpoints.

Extreme Value Theorem

This theorem guarantees 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. It provides the foundational rationale for evaluating the function at critical points and at the endpoints of the interval to find these extrema.

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