Critical Points Calculator
Work out critical points instantly with clear inputs, formula shown and shareable results.
About Critical Points Calculator
Critical points are where the derivative is zero or undefined. For a polynomial they are the roots of f', found here by sign-change bisection, and the second derivative test classifies each as a maximum or minimum.
Formula
Critical point
f'(x) = 0; f''(x) > 0 gives a minimum, f''(x) < 0 a maximum
Worked example
With the default values below, Critical Points Calculator returns Critical points with classification of -1 (maximum), 3 (minimum). Change any input above and every figure here is recalculated in your browser.
Inputs used
- Polynomial coefficients, highest power first
- 1, -3, -9, 5
- Search from x =
- -10
- Search to x =
- 10
Results
- Critical points with classification
- -1 (maximum), 3 (minimum)
- How many found
- 2
- Derivative polynomial
- 3x^2 - 6x - 9
- First critical x
- -1.000000
Step by step
- 1Values used
- 2Critical point
- 3Critical points with classification= -1 (maximum), 3 (minimum)
- 4How many found= 2
- 5Derivative polynomial= 3x^2 - 6x - 9
- 6First critical x= -1.000000
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 − 3x² − 9x + 5?
f' = 3x² − 6x − 9 = 3(x − 3)(x + 1), so the critical points are x = −1 (maximum) and x = 3 (minimum).
What if the second derivative is zero?
The test is inconclusive; check the sign of f' on either side, or use higher derivatives.