How this calculation works
A differentiable polynomial can reach an interval extreme at an endpoint or an interior stationary point. Differentiate, approximate the derivative’s real roots inside the interval and compare the polynomial values. The candidate table exposes the comparison rather than reporting only a graph estimate.
Extrema candidates = both closed-interval endpoints and interior zeros of f′(x)A worked example
For x³−3x on [−2,2], candidates are −2, −1, 1 and 2. Their values are −2, 2, −2 and 2, so the minimum is −2 and maximum 2, each attained at two locations.
Good to know
Real univariate polynomial degree ≤6 on a strictly ordered finite closed interval. Floating-point derivative-root isolation is approximate and may merge close or near-zero candidates. It is not certified global optimization or a proof of exact root multiplicity. One location is reported for tied extrema.
Reference: OpenStax Calculus: extrema and the closed-interval method (opens in a new tab) ↗
A closed interval makes endpoints part of the problem
An interior stationary point is only one type of candidate. A monotone polynomial can reach both its interval extremes at the endpoints, without any interior zero of its derivative. Leaving out the endpoints would miss the answer.
The tool compares a finite list of candidates after differentiating. For x³−3x on [−2,2], the candidate values at −2, −1, 1 and 2 are −2, 2, −2 and 2. Both the minimum and maximum are attained twice in this example.
The numerical polynomial root calculator can inspect one derivative equation with a chosen bracket or starting guess. An interval-extrema candidate list is a different comparison.
Candidate roots are numerical rather than certified
The derivative-recursive search partitions low-degree polynomial intervals at approximate roots of successive derivatives. Sign changes then locate additional candidates, while a scale-relative near-zero test retains potential repeated-root locations.
Close roots, coefficient cancellation and near-zero values can be difficult in floating-point arithmetic. The output therefore labels the minimum and maximum as estimated and exposes the candidate table. It is not a symbolic root-multiplicity certificate or an exact global-optimization proof.
Distinguish an extreme value from its locations
The minimum is a function value; its location is an x coordinate. Several x coordinates can share the same extreme value. This page returns one location and leaves the table available to inspect ties within displayed precision.
For a constant polynomial, every point in the interval is both a minimum and a maximum. No discrete interior list can enumerate that continuum. The numerical polynomial root calculator can separately examine a selected derivative equation, while the tangent calculator describes local slope at a chosen point.
A few questions, answered
Does every derivative zero give an extremum?
No. A stationary point can be an inflection. Comparing function values is still necessary.
Why does the result give only one tied location?
The primary metrics retain one location for each numerical extreme. The candidate table allows comparison of other listed locations.
Can I use this on an open or infinite interval?
No. This calculator is expressly the finite closed-interval polynomial comparison described in its inputs.