Sketch the graph of each function and determine whether the...

Sketch the graph of each function and determine whether the function has any absolute extreme values on its domain. Explain how your answer is consistent with Theorem 1.
$$f(x)=\left\{\begin{array}{lr}
x+1, & -1 \leq x<0 \\
\cos x, & 0<x \leq \frac{\pi}{2}
\end{array}\right.$$

Key Concepts

Piecewise Functions

Piecewise functions are those defined by different expressions over distinct parts of their domain. This concept requires understanding how each piece behaves and how the definitions connect (or do not connect) at the boundaries, which can have implications for continuity, differentiability, and the existence of extrema.

Absolute Extreme Values

Absolute extreme values refer to the absolute (global) minimum or maximum values that a function takes over its entire domain. Finding these values generally involves evaluating the function at critical points and the endpoints of its domain, bearing in mind that discontinuities or open intervals may lead to the absence of absolute extrema.

Extreme Value Theorem

The Extreme Value Theorem states that if a function is continuous on a closed and bounded interval, then it must attain both an absolute minimum and an absolute maximum on that interval. This concept emphasizes the importance of the domain being both closed (includes its endpoints) and bounded, and it provides a criterion for when absolute extrema are guaranteed.

Continuity and Domain Endpoints

Continuity and the consideration of domain endpoints are crucial when analyzing functions for extreme values. If a function is not continuous or if the interval is not closed (i.e., missing endpoints), then the conditions of the Extreme Value Theorem may not hold. This means that even if a function behaves well locally, the lack of a boundary point can result in the non-existence of an absolute maximum or minimum over its domain.

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