Relative Extrema Calculator
Locate local maxima, local minima and inflection points on any interval.
Enter your values
ⓘ Live results
- Critical points found
- 2
- Local maxima
- (1, 5)
- Local minima
- (3, 1)
- Inflection points
- 2
| x | f(x) | f″(x) | Classification |
|---|---|---|---|
| 1 | 5 | -6 | Local maximum |
| 3 | 1 | 6 | Local minimum |
Quick answer
Relative extrema are points where a function changes from increasing to decreasing (local maximum) or decreasing to increasing (local minimum). They occur at critical points where f′(x) = 0 or is undefined.
f′(c) = 0 and f″(c) < 0 → local max; f″(c) > 0 → local min
How to use the Relative Extrema Calculator
- 1Set the function and the x-range to search.
- 2The calculator solves f′(x) = 0 across the range.
- 3Each critical point is classified with the second derivative test.
Worked example
Using the values the calculator loads with:
Inputs
- Function f(x)x^3 - 6x^2 + 9x + 1
- Search from x =-5
- Search to x =8
Results
- Critical points found2
- Local maxima(1, 5)
- Local minima(3, 1)
- Inflection points2
Frequently asked questions
What is the difference between relative and absolute extrema?
A relative (local) extremum is the highest or lowest value in a small neighbourhood. An absolute extremum is the highest or lowest value on the whole domain or interval.
What does the second derivative test tell you?
At a critical point c, f″(c) < 0 means the curve is concave down so c is a local maximum; f″(c) > 0 means concave up, a local minimum; f″(c) = 0 is inconclusive.
About this calculator
- Built and reviewed by James Whitfield · How we build and test calculators
- Report an error
Results are estimates for general information. For medical, legal, structural or financial decisions, confirm with a qualified professional.