Relative Extrema Calculator
Locate local maxima, local minima and inflection points on any interval.
Enter your values
Enter the function f(x) you are working with.
Enter the search from x = you are working with.
Enter the search to x = you are working with.
Results update instantly as you type — no submit needed. Your values are remembered on this device, and the shareable link reopens the calculator with exactly these numbers.
- 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
At a glance
| What it does | Locate local maxima, local minima and inflection points on any interval. |
|---|---|
| Category | Math |
| Inputs needed | Function f(x), Search from x =, Search to x = |
| Main output | Critical points found |
| Formula | f′(c) = 0 and f″(c) < 0 → local max; f″(c) > 0 → local min |
| Cost | Free — no sign-up, no download |
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.
Inputs explained
- Function f(x)
- Enter the function f(x) you are working with.
- Search from x =
- Enter the search from x = you are working with.
- Search to x =
- Enter the search to x = you are working with.
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.
What do I need to enter into the Relative Extrema Calculator?
Just 3 values: function f(x), search from x =, search to x =. Every field starts with a realistic example, so you can change one number at a time and watch the result update instantly.
How does the Relative Extrema Calculator work out the 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. It applies the formula f′(c) = 0 and f″(c) < 0 → local max; f″(c) > 0 → local min and shows the working so you can check each step by hand.
Is the Relative Extrema Calculator free, and do I need an account?
It is completely free with no sign-up, no download and no usage limit. Everything is calculated in your browser, so the numbers you type never leave your device.
Results are estimates for general information. For medical, legal, structural or financial decisions, confirm with a qualified professional.