Inflection Point Calculator
Work out inflection point instantly with clear inputs, formula shown and shareable results.
About Inflection Point Calculator
An inflection point is where the concavity changes, which requires the second derivative to change sign. A zero second derivative alone is not enough; the sign must actually flip.
Formula
Inflection point
f''(x) = 0 with a sign change in f'' on either side
Worked example
With the default values below, Inflection Point Calculator returns Inflection points of x = 2, f = 3. Change any input above and every figure here is recalculated in your browser.
Inputs used
- Polynomial coefficients, highest power first
- 1, -6, 9, 1
- Search from x =
- -10
- Search to x =
- 10
Results
- Inflection points
- x = 2, f = 3
- How many found
- 1
- Second derivative
- 6x - 12
- First inflection x
- 2.000000
Step by step
- 1Values used
- 2Inflection point
- 3Inflection points= x = 2, f = 3
- 4How many found= 1
- 5Second derivative= 6x - 12
- 6First inflection x= 2.000000
Assumptions and limits
- The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 6 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
For x³ − 6x² + 9x + 1?
f'' = 6x − 12, so the inflection point is at x = 2 where f = 3.
Why is f'' = 0 not sufficient?
For x⁴ the second derivative vanishes at 0 but does not change sign, so the curve stays concave up throughout.