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

Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates. $$f(\theta)=\tan \theta, \quad-\frac{\pi}{3} \leq \theta \leq \frac{\pi}{4}$$

Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.
$$f(\theta)=\tan \theta, \quad-\frac{\pi}{3} \leq \theta \leq \frac{\pi}{4}$$

Key Concepts

Extreme Value Theorem

The Extreme Value Theorem states that if a function is continuous on a closed interval, then it must have both an absolute maximum and an absolute minimum value on that interval. This theorem is foundational in calculus because it guarantees the existence of extrema, and it directs the method of evaluating the function at critical points and at the endpoints of the interval to determine these extrema.

Critical Points

Critical points of a function are points in the domain where the derivative is either zero or undefined. They are essential in finding local extrema because these are the only points, aside from endpoints, where the function's rate of change shifts direction. Evaluating the function at these points can help in identifying where absolute maximum or minimum values may occur on the interval.

Endpoint Evaluation

When searching for absolute extrema on a closed interval, it is crucial to evaluate the function at the interval's endpoints as well as at any critical points within the interval. Endpoints are often candidates for absolute maximum or minimum values, especially when the behavior of the function changes near the boundaries of the interval.

Trigonometric Functions

Trigonometric functions, such as the tangent function, have distinct characteristics including periodicity and points of discontinuity (vertical asymptotes). Understanding these properties is important when analyzing and graphing these functions, particularly in determining the interval over which the function is continuous and suitable for applying methods to find extrema.

Graphing Functions

Graphing a function involves plotting its curve on a coordinate plane to visually represent its behavior. This process is valuable for identifying features such as trends, intercepts, and extrema. In the context of finding absolute maximum and minimum values, a graph can help verify analytical findings and provide insight into the function’s overall behavior on the given interval.

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