Section 4.4

Finding Absolute Extrema

The Extreme Value Theorem guarantees that on a closed interval, there must be a highest and lowest point. But where are they? They hide in one of two places: the peaks (critical points) or the edges (endpoints).

The Closed Interval Method

To find the absolute max and min of a continuous function on a closed interval :

1
Find Critical Points
Find where or is undefined inside .
2
Evaluate Endpoints
Calculate and .
3
Compare
The largest value is the Absolute Max.
The smallest value is the Absolute Min.

Why check endpoints? Because the highest point might not be a "peak" (zero slope)—it might just be where the graph was chopped off!

Worked Example

Polynomial on an Interval

Find the absolute extrema of on .

Interval Slider

Left (a): -1

Right (b): 2

Drag the sliders to resize the interval

. Watch how the Max/Min jumps between endpoints and critical points.

Step 1: Critical Points

. Both are in .

Step 2: Evaluate All Candidates

(End):
(Crit):
(Crit):
(End):

Step 3: Conclusion

Absolute Max: 16 (at )
Absolute Min: -1 (at )

Level Up Examples

Example A: Trigonometry

Find absolute extrema of on .

1. Differentiate:

2. Critical Points:

Factor : .

In interval , only .

3. Evaluate Table:

Max
Min

Example B: Root Function

Find extrema for on .