Calculator Inputs
This tool analyzes polynomial functions on a chosen interval. Enter the degree, coefficients, graph range, and calculation precision.
Example Data Table
Sample polynomial: f(x) = x3 - 6x2 + 9x + 1 on the interval [-1, 5].
| x-value | f(x) | f′(x) | f″(x) | Interpretation |
|---|---|---|---|---|
| 1 | 5 | 0 | -6 | Local maximum |
| 3 | 1 | 0 | 6 | Local minimum |
| 2 | 3 | -3 | 0 | Descending between critical points |
Formula Used
1) First derivative test
Critical points of a smooth polynomial occur where f′(x) = 0. The calculator differentiates the polynomial and solves that derivative numerically within the selected interval.
2) Second derivative classification
For each detected point x = c, the calculator evaluates f″(c). If f″(c) > 0, the point is a local minimum. If f″(c) < 0, the point is a local maximum.
3) Flat point review
When the second derivative is also close to zero, the tool checks derivative signs on both sides. This helps distinguish a true extremum from a stationary inflection or other flat point.
4) Interval behavior
After sorting the critical points, the interval is split into subintervals. The sign of f′(x) at each midpoint shows whether the function is increasing or decreasing there.
How to Use This Calculator
- Choose the polynomial degree from 1 to 8.
- Enter each coefficient for the visible powers of x.
- Set the minimum and maximum x-values for the search interval.
- Adjust the sampling points and tolerance for finer detection.
- Press Find Critical Points to generate the result section above the form.
- Review the critical point table, monotonic intervals, and graph.
- Use the CSV or PDF buttons to save the calculated output.
Quick Notes
Polynomial functions only Interval-based search Derivative classification CSV export PDF export Plotly graphFAQs
1) What is a critical point?
A critical point is a point where the derivative becomes zero or undefined. For polynomials, the derivative is always defined, so critical points occur where f′(x) equals zero.
2) Does this tool support any function?
This version is designed for polynomial functions. That keeps the page fully self-contained and allows reliable derivative generation, root scanning, interval analysis, and graphing in one file.
3) Why do I need an interval?
The interval limits the search region. A polynomial may have critical points outside your current graph window, so the chosen x-range controls which ones appear in the results.
4) What does the second derivative tell me?
The second derivative shows curvature near the critical point. Positive curvature suggests a local minimum, while negative curvature suggests a local maximum.
5) Why are sampling points included?
Sampling helps locate promising intervals for the derivative root search. More points improve detection quality, especially when roots are close together or the function changes rapidly.
6) What happens if no critical points are found?
The result table will report that none were detected in your chosen interval. The function may still have critical points outside that range, so widening the interval can help.
7) Can a critical point fail to be a maximum or minimum?
Yes. Some critical points are stationary inflection points or other flat points. The calculator checks derivative behavior around the point to label those cases more accurately.
8) What do the export buttons save?
The CSV export saves the critical point table in spreadsheet-friendly format. The PDF export saves a clean summary with the function, derivatives, and all detected point classifications.