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Using the Candidates Test to Determine Absolute (Global) Extrema

Abigail Young

Abigail Young

7 min read

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Study Guide Overview

This study guide covers finding absolute extrema of a function using the Candidates Test. It explains the steps of the test: finding critical points, evaluating the function at these points and the interval endpoints, and comparing the resulting values. The guide includes a walkthrough example and practice problems with solutions.

5.5 Using the Candidates Test to Determine Absolute (Global) Extrema

In the previous guide, we learned how to find the relative or local extrema of a function using derivatives—if the point is a critical point and the derivative of the function is negative to its left and positive to its right, it is a relative minimum (vice versa it is a relative maximum). Naturally one may wonder, can we use derivatives to find the absolute or global extrema of a function? The answer is yes, we can! 😁

🔎 Finding Absolute Extrema

To find the absolute extrema of a function on a closed interval, one should apply the Candidates Test. Remember from key topic 5.2 that absolute extrema are the maximum and minimum values of a function over its entire domain, rather than a specific interval.

Something to keep in mind for absolute extrema is that they can only occur at critical points or at endpoints of a function on a closed interval.

🪜 Candidates Test Steps

The following steps outline the process for using the Candidates Test to determine absolute extrema:

  1. 🧐 Find the critical points of the function. Remember, the critical points of a function are the points where the first derivative of the function is equal to zero or undefined.
  2. ✏️ Evaluate the function at the critical points. This will give you the y-values of the critical points.
  3. 🔚 Evaluate the function at the endpoints of the interval. This will get you the y-values of the endpoints.
  4. 🍎🍏 Compare the y-values of the critical points and endpoints to determine the absolute extrema. The largest y-value is the absolute maximum, and the smallest y-...