Calculator
Example data table
Example polynomial: f(x) = x^3 − 3x + 1
| Degree | Coefficients (a3, a2, a1, a0) | Expected critical points | Notes |
|---|---|---|---|
| 3 | 1, 0, -3, 1 | 2 real | One local maximum and one local minimum. |
| 4 | 1, 0, -2, 0, 1 | Up to 3 real | Critical points depend on coefficient balance. |
| 5 | 1, 0, 0, -10, 0, 0 | At least 1 real | Odd degree always has real x-intercepts. |
Formula used
A polynomial is f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0.
Turning points occur at critical points where f'(x) = 0. The derivative is f'(x) = n a_n x^{n-1} + (n-1)a_{n-1} x^{n-2} + ... + a_1.
Classification uses the second derivative:
- If f''(x) > 0, the point is a local minimum.
- If f''(x) < 0, the point is a local maximum.
- If f''(x) ≈ 0, the next derivative is checked to flag stationary inflection or higher-order flat points.
How to use this calculator
- Set the polynomial degree from 1 to 10.
- Enter coefficients from the highest power down to the constant.
- Click Compute turning points to solve f'(x)=0.
- Review the table for each critical x, y value, and type.
- Use the interval table to see where the function rises or falls.
- Download the computed table using CSV or PDF buttons.
Numerical methods are used; near-flat cases may require higher precision.
FAQs
1) What is a turning point?
A turning point is a location where the curve changes from increasing to decreasing, or vice versa. It happens at a critical point where the first derivative becomes zero.
2) Are all critical points turning points?
No. Some critical points are stationary inflection points, where the slope is zero but the curve continues to increase or decrease on both sides.
3) Why do you use the second derivative?
The second derivative indicates curvature. Positive curvature suggests a local minimum, while negative curvature suggests a local maximum at the critical point.
4) What if f''(x) is zero at the critical point?
When the second derivative is near zero, the test is inconclusive. The calculator checks higher derivatives to mark stationary inflection or higher-order flat behavior.
5) How many turning points can a polynomial have?
A degree n polynomial can have at most n−1 critical points, and at most n−1 turning points. Some critical points may be non-turning.
6) Why might results look slightly different from hand calculations?
This tool uses numerical root finding. Rounded input coefficients and floating precision can slightly shift computed roots, especially when critical points are very close together.
7) Can this handle repeated roots in the derivative?
Repeated roots can be difficult numerically. The solver attempts to converge and merges very close roots, but extremely ill-conditioned cases may need simpler coefficients or rescaling.
8) What does “increasing/decreasing intervals” mean?
These intervals describe where the derivative stays positive or negative between critical points. Positive derivative means the function increases, and negative derivative means it decreases.