Calculator Input
Enter coefficients for f(x) = ax4 + bx3 + cx2 + dx + e. Lower degree functions also work by setting leading coefficients to zero.
Example Data Table
| Example Function | Critical x Values | f″(x) at Critical Points | Result |
|---|---|---|---|
| f(x) = x4 - 4x2 + 3 | x = -1.4142, 0, 1.4142 | 16, -8, 16 | Minima at ±1.4142, maximum at 0 |
| f(x) = -x4 + 4x2 | x = -1.4142, 0, 1.4142 | -16, 8, -16 | Maxima at ±1.4142, minimum at 0 |
| f(x) = x3 | x = 0 | 0 | Second derivative test is inconclusive |
Formula Used
Polynomial form: f(x) = ax4 + bx3 + cx2 + dx + e
First derivative: f′(x) = 4ax3 + 3bx2 + 2cx + d
Second derivative: f″(x) = 12ax2 + 6bx + 2c
Test: Solve f′(x) = 0. If f″(c) > 0, c is a relative minimum. If f″(c) < 0, c is a relative maximum. If f″(c) = 0, the second derivative test is inconclusive.
This tool calculates all real critical points from the first derivative. It then evaluates the second derivative at each critical point. Positive second derivative values indicate upward curvature. Negative values indicate downward curvature. Zero means you need another method, such as sign analysis or higher derivatives.
How to Use This Calculator
- Enter the coefficients for your polynomial function.
- Use zero for any missing higher degree term.
- Set the graph minimum and maximum x values.
- Choose how many decimal places to display.
- Press Find Relative Extrema to calculate results.
- Read the classification table for maxima, minima, or inconclusive points.
- Review the plot to see curve shape and turning points.
- Use the export buttons to save a CSV or PDF copy.
FAQs
1. What does this calculator find?
It finds real critical points for quartic or lower degree polynomials. Then it uses the second derivative test to classify each point as a relative maximum, relative minimum, or inconclusive point.
2. What is a critical point?
A critical point occurs where the first derivative equals zero. For polynomial functions, these are the x values where the slope becomes horizontal and a turning point may occur.
3. Why can the result be inconclusive?
If the second derivative equals zero at a critical point, this test cannot confirm a maximum or minimum. The function may still change shape there, so another method is needed.
4. Does this work for cubic functions too?
Yes. Enter zero for the quartic coefficient. The tool also works for quadratic, linear, and constant cases, although some functions may produce fewer or no real critical points.
5. Why do graph markers sometimes not appear?
Markers appear only when a critical point lies inside your chosen graph range. Expand the minimum and maximum x values if you want the plot to show more turning points.
6. Are the exported files based on my current result?
Yes. The CSV and PDF buttons export the currently visible results table. Recalculate first if you change any coefficient, graph limit, or display precision.
7. Does the second derivative test always classify extrema?
No. The method works only when the second derivative at the critical point is positive or negative. A zero value leaves the classification unresolved.
8. Can I use decimals and negative coefficients?
Yes. The inputs accept positive, negative, and decimal values. That makes the tool useful for modeled data, class examples, and custom function studies.