| # | x-value | y-value | Tangent Line | Classification |
|---|
Calculator Input
Enter coefficients for a polynomial up to degree 5. The tool solves where the derivative equals zero, then reports horizontal tangent coordinates.
Plotly Graph
Example Data Table
| Example Function | Derivative | Derivative Zeros | Horizontal Tangent Points |
|---|---|---|---|
| f(x) = x³ - 3x | f′(x) = 3x² - 3 | x = -1, 1 | (-1, 2), (1, -2) |
| f(x) = x⁴ - 4x² | f′(x) = 4x³ - 8x | x = -√2, 0, √2 | (-1.414, -4), (0, 0), (1.414, -4) |
| f(x) = x² + 4x + 1 | f′(x) = 2x + 4 | x = -2 | (-2, -3) |
Formula Used
Goal: A horizontal tangent line appears where the slope equals zero.
Condition: f′(x) = 0
Polynomial: f(x) = a₅x⁵ + a₄x⁴ + a₃x³ + a₂x² + a₁x + a₀
Derivative: f′(x) = 5a₅x⁴ + 4a₄x³ + 3a₃x² + 2a₂x + a₁
Tangent line at a horizontal point: y = f(c) where f′(c) = 0
After finding each solution c, the calculator evaluates f(c). It then reports the point (c, f(c)) and the tangent line y = f(c). The second derivative helps classify each point as a local maximum, local minimum, or possible stationary inflection.
How to Use This Calculator
- Enter polynomial coefficients from the x⁵ term down to the constant.
- Set graph minimum, maximum, and sample count.
- Click Calculate Horizontal Tangents.
- Review the result summary above the form area.
- Check the table for x-values, y-values, and classifications.
- Inspect the graph to see tangent points on the curve.
- Download the computed table as CSV or PDF.
Frequently Asked Questions
1. What is a horizontal tangent line?
A horizontal tangent line touches a curve where its instantaneous slope equals zero. At that point, the tangent line is flat and parallel to the x-axis.
2. Does every derivative zero create a maximum or minimum?
No. Some derivative zeros are stationary inflection points. The graph may flatten without switching direction. The second derivative or sign changes help classify the point.
3. What functions does this calculator support?
This page supports polynomial functions up to degree five. You enter coefficients directly, and the calculator builds the function automatically.
4. Why do I sometimes get no horizontal tangent points?
If the derivative has no real solutions, the curve has no real horizontal tangents. Some functions only have complex derivative roots, which do not appear on the real graph.
5. Why is the tangent line written as y = constant?
At a horizontal tangent point, the slope is zero. The tangent line therefore has the form y = f(c), which is a constant-height horizontal line.
6. How accurate are the root values?
The tool uses numerical scanning and refinement. Results are highly practical for study and graph analysis, though symbolic algebra systems may show exact radical forms.
7. Can repeated roots appear in the result?
Repeated derivative roots can occur. The calculator merges nearby duplicates, then reports a single critical x-value for cleaner output and interpretation.
8. What should I choose for graph limits?
Pick limits wide enough to show turning behavior around critical points. If important features seem missing, expand the x-range and recalculate the graph.