Calculator
Example Data Table
| Function | x0 | f'(x0) | f''(x0) | Classification |
|---|---|---|---|---|
| x² | 0 | 0 | 2 | Local Minimum |
| -x² | 0 | 0 | -2 | Local Maximum |
| x³ | 0 | 0 | 0 | Inconclusive |
| x² + 4x + 4 | -2 | 0 | 2 | Local Minimum |
Formula Used
The second derivative test is applied at a stationary point x0. First, check whether f'(x0) = 0. If the point is stationary, use the sign of the second derivative to classify the local behavior.
- If f'(x0) = 0 and f''(x0) > 0, the point is a local minimum.
- If f'(x0) = 0 and f''(x0) < 0, the point is a local maximum.
- If f'(x0) = 0 and f''(x0) = 0, the test is inconclusive.
This calculator also builds a local quadratic model near the point:
L(x) = f(x0) + f'(x0)(x - x0) + 0.5f''(x0)(x - x0)²
That model supports the neighborhood table and the graph shown after calculation.
How to Use This Calculator
- Enter a function label or short note for reference.
- Enter the candidate point x0 where you want to test local behavior.
- Provide f(x0), f'(x0), and f''(x0) at that point.
- Set a tolerance to decide when the first derivative should be treated as zero.
- Choose a graph radius and decimal precision.
- Press the calculate button to view the classification under the header.
- Review the summary, neighborhood table, and Plotly graph.
- Use the CSV or PDF buttons to save the result.
Why This Test Matters
The second derivative test is a fast method for checking whether a stationary point behaves like a local maximum or local minimum. It is useful in calculus classes, optimization tasks, and function analysis exercises.
Because the test depends on local curvature, it gives quick insight into the shape of the graph near a candidate point. When the result is inconclusive, that is also valuable because it tells you more analysis is needed.
This page is designed for direct value entry. It works well when derivative values have already been found by hand, from symbolic work, or from another system.
FAQs
1. What does the second derivative test determine?
It checks whether a stationary point is a local maximum, local minimum, or inconclusive by using the sign of the second derivative at that point.
2. When can I use this calculator?
Use it after you already know the candidate point and the values of f(x0), f'(x0), and f''(x0). It is ideal for calculus homework and quick verification.
3. What if f'(x0) is not zero?
The point is not stationary under the chosen tolerance, so the standard second derivative test does not confirm it as a local extremum.
4. What if f''(x0) equals zero?
The test becomes inconclusive. The point may still be a maximum, minimum, or neither, so you should use other methods such as sign analysis.
5. Why is a tolerance field included?
Real calculations often produce tiny rounding errors. The tolerance lets you treat very small first derivative values as zero when appropriate.
6. What does the graph represent?
The graph shows a local quadratic model built from the entered values. It helps visualize the nearby shape around the tested point.
7. Can a critical point fail to be a local extremum?
Yes. Some critical points are saddle points, inflection points, or flat points. That is why the inconclusive outcome is important in calculus.
8. Does a positive second derivative always mean a minimum?
Only at a stationary point. A positive second derivative alone indicates upward curvature, but the local minimum conclusion requires f'(x0) to be zero.