Description
Apply the two-variable Hessian determinant test to supplied entries.
Derivative tests classify local behavior only under their hypotheses, including a stationary point when required. An inconclusive or flat-within-tolerance result is distinct from a proved extremum; supplied derivatives and a chosen tolerance do not establish smoothness or a global optimum.
Apply the two-variable Hessian determinant test to supplied entries.
When to use this tool
Use Critical Point Second Derivative Discriminant when checking the stated derivative-test or local stationary-point convention.
How to use the tool and how it works
- Enter xx, xy, yy. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read determinant, classification. Use the output convention below.
Apply the two-variable Hessian determinant test to supplied entries. determinant = xx*yy-xy²; negative gives saddle, zero inconclusive, positive gives minimum when xx > 0 and maximum otherwise.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| xx | Xx | Required. Finite number. |
| xy | Xy | Required. Finite number. |
| yy | Yy | Required. Finite number. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| determinant | Determinant | Required. Finite number. |
| classification | Local Hessian test label; stationarity and a suitable smooth symmetric Hessian must be established separately. | Required. Text; maximum UTF-16 length 8192. |
{
"input": {
"xx": 2,
"xy": 1,
"yy": 3
},
"output": {
"determinant": 5,
"classification": "minimum"
}
}Limitations and assumptions
- This assumes an independently established stationary point and a valid symmetric Hessian under sufficient smoothness; none is verified by these inputs.
- Inputs must satisfy both the field types and the cross-field conditions above. Invalid input causes an error; a successful false flag or documented placeholder is a different result.
- Numeric calculations use binary floating point unless exact decimal-integer arithmetic is explicitly stated. No hidden measurement tolerance or decimal rounding is added; nonfinite or out-of-contract results are rejected. Numeric signed-zero results are exported as zero.
Alternative or Complementary methods
Verify the actual function, stationarity and smoothness assumptions, then use the relevant complete local or global analysis to establish an extremum.
References
Similar or alternative tools
- Calculus: Stationary Point Second Derivative Sign
Classify a supplied stationary point by its second derivative.
- Scalar Critical Point Flatness Residual
Locate the first derivative outside a supplied flatness tolerance.