In Exercises 15-20, sketch the graph of each function and...

In Exercises $15-20,$ 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 $1 .$ $$ f(x)=\left\{\begin{array}{ll}{x+1,} & {-1 \leq x<0} \\ {\cos x,} & {0 < x \leq \frac{\pi}{2}}\end{array}\right. $$

In Exercises $15-20,$ 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 $1 .$
$$
f(x)=\left\{\begin{array}{ll}{x+1,} & {-1 \leq x<0} \\ {\cos x,} & {0 < x \leq \frac{\pi}{2}}\end{array}\right.
$$

Key Concepts

Piecewise Functions

A piecewise function is defined by different expressions over different parts of its domain. In assessing such functions, it is important to check the behavior and continuity at the boundary points where the defining formulas change, ensuring that the transitions are well?understood and that the overall function is properly interpreted on its entire domain.

Continuity on Closed Intervals

For functions defined on a closed interval, verifying continuity across the entire interval—including at the junction points of a piecewise definition—is crucial. If a function is continuous throughout a closed domain, it adheres to properties that guarantee the existence of absolute (global) extreme values, as discontinuities could potentially prevent extreme values from being attained.

Absolute Extreme Values

Absolute extreme values refer to the highest and lowest function values over its entire domain. To determine these, one examines not only any critical points in the interior of the domain but also the endpoint values, since maxima or minima can occur at these boundaries. This comprehensive check is essential especially in the context of piecewise functions where different formulas may behave differently over various subintervals.

Extreme Value Theorem

The Extreme Value Theorem states that if a function is continuous on a closed interval, then it must attain both a maximum and a minimum value on that interval. This theorem provides a theoretical basis for expecting to find absolute extrema, and it guides the examination of the function by ensuring that, under the right conditions, the function will have both an absolute highest and lowest point within the given domain.

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