Finding the Best
One of the most useful things derivatives do is optimisation: finding where a quantity is largest or smallest. Because the tangent is horizontal at a peak or a valley, the derivative is zero at these interior extreme points.
To find these, we locate critical points: values where or where the derivative does not exist. On a closed interval, you must also check the endpoints, since the extreme value can occur there even when the derivative is not zero.
| Test Type | Condition | Conclusion |
|---|---|---|
| First Derivative | changes to | Local maximum |
| First Derivative | changes to | Local minimum |
| Second Derivative | ( measures concavity) | Concave up (minimum) |
| Second Derivative | Concave down (maximum) |
Tests & Real-World Use
A zero derivative only marks a critical point, not automatically a peak or valley. For instance, at has but is an inflection point (where changes sign), which is neither a maximum nor a minimum.
| Common Pitfall | The Reality |
|---|---|
| Assuming means max/min | It could be an inflection point; you must classify it. |
| Ignoring endpoints | On closed intervals, extremes often sit at the edges where . |
This mathematics drives real-world computing: training a machine-learning model minimises a loss function, and resource allocation maximises utility. Gradient descent follows the derivative downhill to find that minimum.