Search an interval for points where the numerical derivative is zero. Formula and critical x-values remain visible in one place for independent verification.
Critical numbers occur where f′ is zero or undefined within the function domain; this numerical search locates zero-derivative candidates.
In critical point, the relationship among the quantities matters just as much as their arithmetic.
Scan derivative signs, bracket crossings, and refine each bracket by bisection.
For x³−3x, derivative 3x²−3 vanishes at x=−1 and x=1. This Critical Point example can be compared with classify candidates.
They organize optimization, monotonicity, and graph shape.
A scan may miss repeated roots, sharp nondifferentiable points, or roots packed more closely than its grid.
An impossible critical x-values sign or magnitude usually points to field assignment before it points to rounding.
The input labels—Function f(x), Interval start and Interval end—encode the model used on this page. Write those labels beside source values when transferring a problem from paper or a spreadsheet. That small step catches transposed quantities and mixed units before they become a polished-looking critical x-values.
Widening the interval can reveal more candidates without changing existing ones. A one-field trial makes this relationship visible without reworking the entire example.
Choose rounding after considering how the result will be used. Comparison may need only a few significant digits, while a later multi-step calculation benefits from carrying more. In either case, retain the page's formula with the critical x-values so the underlying definition remains visible.
Local extrema classify critical candidates by surrounding derivative behavior. The distinction determines whether this page fits the original question.
If the number moves into a spreadsheet, give its cell a critical x-values heading and retain the source values Function f(x), Interval start and Interval end nearby. Context matters more than extra displayed digits.
A reproducible Critical Point result needs Function f(x), Interval start, and Interval end. Inspect the function at every stated bound and at several interior points. Undefined values, steep gradients, and oscillation may require splitting the interval or choosing a different method even when the calculator can sample most points successfully.
For this critical point result, report whether the answer is a numerical estimate or an exact algebraic result. Also retain any step size, panel count, direction, or iteration limit that materially shaped the displayed digits.
No.
Endpoint treatment depends on the problem.
Yes.