Finding Absolute and Local Extrema
Simple Explanation
A local extremum is a peak or valley relative to nearby points; an absolute extremum is the single largest or smallest value over an entire interval (or the whole domain). On a closed interval [a,b], the absolute extrema always occur either at a critical point or at one of the two endpoints.
Why Do We Need It?
Absolute extrema answer real optimization questions directly — the biggest possible profit, the smallest possible cost, the maximum height — rather than just describing local shape.
See It
A cubic curve on a bounded interval, with its local max, local min, and both endpoints marked, showing the highest and lowest points overall
Formula
Classifying Extrema (First Derivative Test)
f' changes + to − at c ⟹ local max at c; f' changes − to + at c ⟹ local min at c
Examining how the derivative's sign changes on either side of a critical point classifies it as a local maximum, local minimum, or neither.
- c
- — a critical point being classified
When to use it: Whenever a critical point needs to be classified as a local max, local min, or neither — and whenever absolute extrema on a closed interval are needed (also checking the endpoints).
Worked Example
Find the absolute extrema on a closed interval
Find the absolute maximum and minimum of f(x) = x³ − 3x on [−2, 2].
Why Does This Work?
On a closed, bounded interval, a continuous function is guaranteed to attain both an absolute maximum and an absolute minimum (the Extreme Value Theorem). That extreme value must happen either where the function momentarily flattens out (a critical point) or at the very edge of the allowed range (an endpoint) — nowhere else is possible, so checking just those candidates is guaranteed to find it.
Real-Life Example
Maximum profit within a limited production range
A factory can only realistically produce between 0 and 500 units per day; profit as a function of units produced has its own shape within that range.
The maximum achievable profit is found the same way — check the critical points of the profit function and the two endpoints (0 and 500 units), then compare.
Practice
For f(x) = x³ − 3x on [−2, 2], what is the absolute maximum value?
HardOn a closed interval [a,b], where can the absolute extrema of a continuous function occur?
MediumCommon mistake
Forgetting to check the endpoints — on a closed interval, an absolute extremum can occur at an endpoint even when the endpoint is not a critical point at all.
Quick Review
- Local extrema: peaks/valleys relative to nearby points, always at critical points.
- Absolute extrema on [a,b]: the single largest/smallest value overall — check critical points AND endpoints.
- Compare all candidate values; the largest is the absolute max, the smallest is the absolute min.