Sketch the graph of f by hand and use your sketch to find...

Sketch the graph of $f$ by hand and use your sketch to find the absolute and local maximum and minimum values of $f .$ (Use the graphs and transformations of Sections 1.2 and $1.3 .$ )
$\mathrm{f}(\mathrm{t})=\cos t, \quad-3 \pi / 2 \leqslant \mathrm{t} \leqslant 3 \pi / 2$

Key Concepts

Cosine Function

The cosine function is a fundamental trigonometric function characterized by its smooth, periodic oscillations between -1 and 1. Understanding its properties, such as periodicity, symmetry, amplitude, and phase, is essential for graphing and analyzing its behavior over any interval.

Graphing and Sketching Functions

Sketching the graph of a function involves understanding its basic shape, key points such as intercepts, peaks, and troughs, and any transformations that may shift or scale the graph. This visual representation aids in comprehending the function's behavior and identifying areas where extrema occur.

Extreme Value Theorem

The Extreme Value Theorem states that if a function is continuous on a closed interval, then it must attain both an absolute maximum and an absolute minimum on that interval. This concept is crucial when analyzing functions over bounded domains, as it ensures that the search for extreme values is well-founded.

Local vs Absolute Extrema

Local (or relative) extrema refer to points where a function reaches a maximum or minimum within a small surrounding interval, while absolute (or global) extrema are the highest or lowest points over the entire domain under consideration. Differentiating between these types of extrema is critical in optimization problems and function analysis.

Domain and Interval Analysis

Analyzing the domain of a function, especially when it is restricted to a specific interval, is vital for understanding its behavior and determining where important features such as extrema may occur. Evaluating the function at endpoints as well as within the interior helps ensure that all potential extreme values are identified.

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