Local vs global?
Local compares nearby points; global is the absolute extreme on a set.
Local Maximum and Minimum Calculator
Solve f′=0 for a quadratic or cubic, then use f″ to classify local max/min.
Enter the polynomial
Choose quadratic or cubic and enter coefficients.
About this calculator
How to use Local Maximum and Minimum Calculator, plus the formulas and logic behind the results.
Tool description
Local extrema occur at critical points. The second derivative test classifies them when f″ ≠ 0.
How to use
- Select quadratic or cubic mode.
- Enter coefficients.
- Select Calculate to list critical points and classifications.
- If f″ = 0, the test is inconclusive—inspect further.
Formulas & reference
Critical: f′(x) = 0f″(x) > 0 → local minimumf″(x) < 0 → local maximumf″(x) = 0 → inconclusive
Extrema tips
- Quadratic with a > 0 has a global minimum at the vertex
- Cubics may have a local max and a local min
- Inconclusive cases need higher derivatives or a sign chart
- Always report the y-value f(x) with the classification
Calculation logic
Finds real roots of f′, evaluates f and f″ there, and labels max/min/inconclusive.
Does not search endpoints of an interval—only local interior extrema candidates.
Local Maximum and Minimum Calculator FAQ
Common questions about this calculator and how to use the results.
What if there is no critical point?
The polynomial may be strictly mono-tonic (e.g. linear).
Quadratic always one critical point?
Yes when a ≠ 0 (the vertex).
When is the test inconclusive?
When f″(c) = 0 at a critical point c.
Are saddle points included?
In 1D, inconclusive flat points may be neither max nor min.
Is this free?
Yes.