Enter polynomial and interval settings
Example data table
Sample polynomial: f(x) = x3 - 6x2 + 9x + 2 on the interval [0, 4].
| x | f(x) | Comment |
|---|---|---|
| 0 | 2 | Left endpoint |
| 1 | 6 | Critical point and local maximum |
| 2 | 4 | Interior sample value |
| 3 | 2 | Critical point and local minimum |
| 4 | 6 | Right endpoint |
Formula used
1. Polynomial evaluation: f(x) = anxn + an-1xn-1 + ... + a1x + a0
2. Range idea on a closed interval [a, b]: evaluate f(a), f(b), and every interior critical point where f'(x) = 0.
3. Derivative rule: if f(x) = anxn, then f'(x) = n anxn-1.
4. Numerical search: this page scans the interval, estimates derivative roots, checks endpoints, then compares candidate y-values to estimate the minimum and maximum.
How to use this calculator
- Enter coefficients from highest degree to the constant term.
- Set the interval start and end where you want the range.
- Choose scan points for analysis depth and graph points for plotting smoothness.
- Set the decimal precision for displayed values.
- Press the calculate button to show the result above the form.
- Review the graph, candidate table, monotonic intervals, and approximate roots.
- Use the CSV button for spreadsheet work or the PDF button for sharing.
Frequently asked questions
1. What does the polynomial range mean?
It is the set of output values produced by the polynomial over your chosen x-interval. This tool reports the estimated minimum and maximum values on that interval.
2. Why does the calculator ask for coefficients?
Coefficients define the polynomial exactly. Enter them in descending power order, such as 2, -3, 0, 5 for 2x³ - 3x² + 5.
3. Are the minimum and maximum exact?
They are numerically estimated from endpoints, detected critical points, and dense interval sampling. Increasing scan points can improve accuracy for difficult polynomials.
4. What are critical points in this tool?
Critical points are interior x-values where the derivative is zero. They are important because closed-interval extrema often occur at endpoints or critical points.
5. Why are roots shown if this is a range calculator?
Roots help you understand where the graph crosses the x-axis. They add context when interpreting the plotted curve and interval behavior.
6. How many scan points should I use?
For most classroom and business use, 500 to 1500 scan points works well. Use more for wide intervals or high-degree polynomials.
7. Can this handle constant or linear polynomials?
Yes. Constant functions return one repeated value, while linear functions usually reach their extrema at the interval endpoints.
8. What do the CSV and PDF downloads include?
The CSV export contains summary metrics, critical points, roots, and monotonic intervals. The PDF export captures the visible result section for quick sharing.