Concavity Calculator
Work out concavity instantly with clear inputs, formula shown and shareable results.
About Concavity Calculator
Concavity is the sign of the second derivative. Positive means the curve bends upward like a bowl holding water; negative means it bends downward. The tangent line lies below a concave-up curve and above a concave-down one.
Formula
Concavity
f''(x) > 0 concave up; f''(x) < 0 concave down
Worked example
With the default values below, Concavity Calculator returns Concavity at your x of Concave up (curving upward). Change any input above and every figure here is recalculated in your browser.
Inputs used
- Polynomial coefficients, highest power first
- 1, -3, 0, 2
- Test at x =
- 2
- Also test at x =
- -2
Results
- Concavity at your x
- Concave up (curving upward)
- f''(x)
- 6.00000000
- Concavity at the second point
- Concave down (curving downward)
- Second derivative
- 6x - 6
Step by step
- 1Values used
- 2Concavity
- 3Concavity at your x= Concave up (curving upward)
- 4f''(x)= 6.00000000
- 5Concavity at the second point= Concave down (curving downward)
- 6Second derivative= 6x - 6
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 8 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 + 2 at x = 2 and x = −2?
f'' = 6x − 6, giving 6 at x = 2 (concave up) and −18 at x = −2 (concave down).
Does concavity tell me about maxima?
Yes. At a critical point, concave up means a minimum and concave down means a maximum.