Calculator Inputs
Example Data Table
| Example function | Critical point | Type | Key note |
|---|---|---|---|
| f(x) = x³ - 6x² + 9x + 15 | x = 1 | Local maximum | Second derivative is negative. |
| f(x) = x³ - 6x² + 9x + 15 | x = 3 | Local minimum | Second derivative is positive. |
| f(x) = -2x² + 8x - 1 | x = 2 | Maximum | Quadratic opens downward. |
Formula Used
Function model: f(x) = ax³ + bx² + cx + d
First derivative: f′(x) = 3ax² + 2bx + c
Second derivative: f″(x) = 6ax + 2b
Critical points: Solve f′(x) = 0 to get stationary values.
Classification rule: If f″(x) < 0, the point is a local maximum. If f″(x) > 0, the point is a local minimum.
Closed interval extrema: Compare function values at critical points and interval endpoints to locate overall maximum and minimum.
How to Use This Calculator
- Enter coefficients a, b, c, and d for your polynomial.
- Set the interval start and end for closed interval analysis.
- Choose a sample step to build the supporting data table.
- Enter a specific x value for direct function evaluation.
- Press Submit to show extrema above the form.
- Review critical points, interval extrema, and the generated derivative table.
- Use CSV download for spreadsheets and PDF download for printable study sheets.
Frequently Asked Questions
1. What does this calculator find?
It finds stationary points, classifies them as maxima or minima, checks interval extrema, evaluates derivatives, and builds a supporting sample table.
2. Can it analyze quadratic functions too?
Yes. Set the cubic coefficient to zero. The calculator then treats the expression as a quadratic and classifies its single turning point.
3. Why are there no critical points sometimes?
Some functions have derivatives without real roots. In that case, the function keeps increasing, decreasing, or stays constant across real x values.
4. What is the difference between local and interval extrema?
Local extrema occur near stationary points. Interval extrema compare those points with endpoints, so they give the overall highest and lowest values on the chosen interval.
5. How does the second derivative help?
A negative second derivative indicates downward curvature, which signals a local maximum. A positive second derivative indicates upward curvature, which signals a local minimum.
6. What is the inflection point?
For cubic functions, the inflection point is where concavity changes. It occurs when the second derivative equals zero.
7. Why include a sample step field?
The sample step controls the spacing of x values in the output table. Smaller steps create denser tables for closer inspection.
8. What does the CSV export contain?
The export includes the function summary, interval extrema, evaluation point values, and can be extended easily for larger worksheet workflows.