Ch. 4 - Applications of the Derivative

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 4 - Applications of the Derivative

Problem 4.1.13

Chapter 4, Problem 4.1.13

Use the following graphs to identify the points (if any) on the interval [a, b] at which the function has an absolute maximum or an absolute minimum value <IMAGE>

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Examine the graph of the function over the interval [a, b]. Identify any endpoints and critical points within this interval.

Determine the function values at the endpoints of the interval, a and b. These are potential candidates for absolute extrema.

Identify any critical points within the interval [a, b]. A critical point occurs where the derivative is zero or undefined. Check the graph for any such points.

Evaluate the function at each critical point identified in the previous step. These values are also candidates for absolute extrema.

Compare the function values obtained at the endpoints and critical points. The largest value is the absolute maximum, and the smallest value is the absolute minimum on the interval [a, b].

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Absolute Maximum and Minimum

An absolute maximum of a function on a given interval is the highest value the function attains within that interval, while an absolute minimum is the lowest value. These extrema can occur at critical points, where the derivative is zero or undefined, or at the endpoints of the interval. Identifying these points is crucial for determining the overall behavior of the function.

Critical Points

Critical points are values in the domain of a function where the derivative is either zero or does not exist. These points are significant because they are potential locations for local maxima and minima. To find absolute extrema, one must evaluate the function at these critical points as well as at the endpoints of the interval.

Evaluating Functions on an Interval

To find absolute extrema, one must evaluate the function at all critical points and at the endpoints of the specified interval. This involves substituting these values into the function to determine which yields the highest and lowest outputs. This process ensures that all possible candidates for absolute extrema are considered.

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