#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-...