How the Critical Point rule is built
Critical numbers occur where f′ is zero or undefined within the function domain; this numerical search locates zero-derivative candidates.
Math calculator
Critical Point Calculator
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 Point inputs
Calculation inputs
Working out Critical Point by hand
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.
Problems suited to Critical Point
They organize optimization, monotonicity, and graph shape.
Reading the Critical Point result
A scan may miss repeated roots, sharp nondifferentiable points, or roots packed more closely than its grid.
Documenting Critical Point for the next step
From Critical Point output to working record
Link Function f(x) to its Critical Point role. Link Interval start to its Critical Point role. The retained Critical Point formula identifies the Critical Point model.
What to preserve with the Critical Point result
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.
Boundary checks for Critical Point
Substitute the Critical Point solution into Function f(x). This Critical Point check rejects false Critical Point branches and forbidden denominators.
Choose an easy Function f(x) value before running Critical Point. Predict Interval start, then compare it with the Critical Point output.
The relationship between Function f(x) and Critical x-values provides another check on Critical Point. Move Function f(x) slightly while keeping Interval start constant, then decide in advance whether Critical x-values ought to rise, fall, or remain unchanged.
Validating Critical Point
Start the Critical Point cross-check with Function f(x). Apply the original Interval start and recompute Critical x-values. Treat a different Interval end as new Critical Point data. That prevents its Critical x-values from being attributed to the earlier Critical Point setup.
Test a boundary relevant to Function f(x) during the Critical Point review. Hold Interval start constant and anticipate Critical x-values. The boundary case can reveal a branch or limit hidden by the original Critical Point numbers.
Questions about Critical Point
Is every critical point an extremum?
No.
Can endpoints be critical?
Endpoint treatment depends on the problem.
Can numerical scanning miss points?
Yes.