Calculator Inputs
Enter a polynomial up to degree 4. The page evaluates f(x), computes f''(x), detects real roots of the second derivative, and classifies each interval.
Formula Used
For the quartic model used here:
f(x) = ax4 + bx3 + cx2 + dx + e
f'(x) = 4ax3 + 3bx2 + 2cx + d
f''(x) = 12ax2 + 6bx + 2c
How to Use This Calculator
- Enter the coefficients for the polynomial, from x4 through the constant term.
- Choose the domain you want to inspect for concavity behavior.
- Set the sample count for the graph and the decimal precision for tables.
- Press Calculate Concavity to show results under the header and above the form.
- Read the interval table, then inspect second-derivative roots and inflection decisions.
- Use the chart and sample table to confirm how the curvature changes visually.
- Export the findings with CSV or PDF when you need documentation.
Example Data Table
Example input: f(x) = x4 - 4x3 + 3x + 1 on the domain [-5, 5].
| Item | Value |
|---|---|
| Second derivative | 12x2 - 24x |
| Roots of f''(x) | x = 0 and x = 2 |
| Concave up intervals | (-5, 0) and (2, 5) |
| Concave down interval | (0, 2) |
| Inflection points | (0, 1) and (2, -13) |
Frequently Asked Questions
1) What does concavity mean?
Concavity describes whether a graph bends upward or downward over an interval. It helps explain shape, curvature, and where a function changes from cup-like to cap-like behavior.
2) Why does the calculator use the second derivative?
The second derivative measures how the slope itself changes. A positive second derivative signals concave up behavior, while a negative value signals concave down behavior.
3) Is every root of f''(x) an inflection point?
No. A root of the second derivative is only an inflection point when the sign of f''(x) actually changes across that x-value.
4) Which functions can this version analyze?
This page analyzes polynomial functions up to degree four. Lower-degree polynomials work too, as long as you enter zero for any missing higher-order coefficients.
5) Why do I need a domain?
A domain limits the analysis to the interval you care about. Roots or curvature changes outside that interval do not affect the reported results.
6) What happens when f''(x) is always zero?
If the second derivative is zero throughout the selected domain, the function has no upward or downward curvature there. Linear functions are the most common example.
7) Does the graph determine the answer?
The graph supports interpretation, but the interval table and sign test provide the actual decision. Visual inspection alone can be misleading on stretched axes.
8) Can I export the results?
Yes. After calculation, use the CSV button for spreadsheet-friendly output or the PDF button for a printable report of the visible results.