The First Derivative Test and the Extreme Values of Functions

Extreme values of a function are the maximum or minimum values that the function attains. They can be:

• **Absolute (or Global) Extrema:** The single highest and lowest values of the function over its entire domain or a specified interval.
• **Local (or Relative) Extrema:** The highest or lowest values of the function within a specific "neighborhood" or open interval around a point. A point is a local maximum if the function value there is greater than or equal to the values at all nearby points; similarly for a local minimum.

The First Derivative Test is used to determine the local maximum and local minimum values of a function. It does this by analyzing the sign of the function's first derivative around its critical points (points where the derivative is zero or undefined).

The test is based on the fact that the first derivative tells us whether a function is increasing or decreasing:

• If f'(x) > 0 on an interval, the function f(x) is increasing on that interval.
• If f'(x) < 0 on an interval, the function f(x) is decreasing on that interval.

At a local extremum (a peak or valley), the function typically changes from increasing to decreasing (local maximum) or decreasing to increasing (local minimum). This change occurs at a critical point.

Follow these steps:

1. Find the first derivative, f'(x).
2. Find the critical points by setting f'(x) = 0 or finding points where f'(x) is undefined (but the original function f(x) is defined).
3. Create a sign chart for f'(x), using the critical points (and any points where the original function is undefined) to divide the number line into intervals.
4. Choose a test value within each interval and plug it into f'(x) to determine the sign (+ or -) of the derivative in that interval.
5. Apply the First Derivative Test conclusion:
   • If the sign of f'(x) changes from **positive to negative** as you move from left to right across a critical point 'c', then f(c) is a **local maximum**.
   • If the sign of f'(x) changes from **negative to positive** as you move from left to right across a critical point 'c', then f(c) is a **local minimum**.
   • If the sign of f'(x) does **not change** across a critical point, then 'c' is neither a local maximum nor a local minimum (it might be an inflection point or just a point where the graph flattens out).
6. Evaluate the original function f(x) at the critical points identified as local extrema to find the actual local maximum or minimum *values*.

To find the absolute extreme values of a continuous function f(x) on a closed interval [a, b], use the **Extreme Value Theorem** and these steps:

1. Verify that the function is continuous on the closed interval [a, b].
2. Find the critical points of f(x) within the open interval (a, b). (These are points where f'(x) = 0 or f'(x) is undefined within the interval).
3. Evaluate the original function, f(x), at the critical points found in step 2.
4. Evaluate the original function, f(x), at the endpoints of the interval, f(a) and f(b).
5. Compare all the values found in steps 3 and 4. The largest value is the absolute maximum, and the smallest value is the absolute minimum on the interval [a, b].

The First Derivative Test helps you find the critical points in step 2, which are candidates for local extrema, but for absolute extrema on a closed interval, you must also check the endpoints.