Optimization (1D min/max finder)

Inspectable records

Ledger state

The current ledger marks this instrument verified clean within the source-bound verification corpus.

Area

Mathematics — arithmetic through advanced (algebra, matrices, calculus, sequences/series, number theory, combinatorics, graphing, complex numbers, geometry, logic, sets)

Tranche

tranche-8

Evidence ID

t8-024

Ledger rows

1

No ledger note is attached.

Review findings

The stated derivative condition and second-derivative test are standard but incomplete. The exact ambiguity requiring review is whether the instrument promises all local extrema or only an extremum of a unimodal objective on a finite interval, and which numerical tolerances, endpoint rules, differentiability assumptions, and minimization/maximization mode apply.

Formula and scope findings

• The stationary-point condition f'(x)=0 finds only differentiable interior candidates; the algorithm must also test interval endpoints and nondifferentiable points such as cusps.
• The second-derivative test is inconclusive when f''(x)=0 and cannot by itself classify inflection points; classification should use a first-derivative sign change, higher derivatives, or a neighborhood test.
• Golden-section search is valid for a continuous unimodal objective on a bounded interval, not for arbitrary functions or for finding all local extrema. The implementation must specify minimization versus maximization and the unimodality assumption.
• A derivative root finder can miss roots without a bracketing/root-isolation policy and requires explicit numerical tolerances and maximum-iteration limits.
• The claim to classify inflection points is underspecified: an inflection point requires a concavity change, not merely a stationary point.

Unit assumptions

• x is a one-dimensional scalar with a consistent unit throughout the function and interval; f(x) has its own consistent output unit.
• The derivative has units of f-units per x-unit, and the second derivative has units of f-units per x-unit squared.
• The search interval is assumed finite, closed, and ordered as [a,b]; endpoint inclusion should be documented.
• Numerical tolerances are required for x, function values, derivative roots, and classification, but none are specified in the instrument spec.

Boundary hardening

BREAK_CLAIMED · SELF_FLAGGED

The specification exposes invalid-input hazards: a zero-width interval and an interval containing a pole must be rejected. It also requires explicit handling of nondifferentiable branch selection and angle-unit conventions; otherwise the returned extrema can be absent, undefined, or numerically wrong.

Errata and corrections

No applied erratum is attached to this unique-name record.

Inspectable example

4 review vectors are recorded. One source-bound sample is shown below.

{
  "inputs": {
    "function": "f(x)=(x-2)^2+1",
    "search_interval": [
      0,
      4
    ],
    "method": "derivative_root_finding"
  },
  "expected_outputs": {
    "derivative": "f'(x)=2x-4",
    "stationary_points": [
      {
        "x": "2",
        "f_x": "1",
        "classification": "local and global minimum",
        "second_derivative": "2"
      }
    ],
    "endpoints": [
      {
        "x": "0",
        "f_x": "5"
      },
      {
        "x": "4",
        "f_x": "5"
      }
    ],
    "spot_checks": [
      {
        "at": 0,
        "input_side": 5,
        "answer_side": 5
      },
      {
        "at": 1.25,
        "input_side": 2.5625,
        "answer_side": 2.5625
      },
      {
        "at": 3.5,
        "input_side": 3.25,
        "answer_side": 3.25
      }
    ]
  },
  "reference_data": {
    "canonical_result": "f'(x)=0 gives 2x-4=0, hence x=2; f''(x)=2>0, so x=2 is a minimum; f(2)=1."
  },
  "working": "1. Differentiate: f'(x)=2(x-2)=2x-4. 2. Set the derivative to zero: 2x-4=0, so 2x=4 and x=2. 3. Evaluate the second derivative: f''(x)=2, and 2>0, so the stationary point is a local minimum. 4. Evaluate the function: f(2)=(2-2)^2+1=0+1=1. 5. Check endpoints: f(0)=(0-2)^2+1=4+1=5; f(4)=(4-2)^2+1=4+1=5. Therefore the reported minimum is (2,1)."
}

Related workflows

No source-defined multi-instrument workflow membership is recorded.

Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

• VERIFIED_LEDGER.json · VERIFIED_CLEAN
• evidence-tranche8/tranche8-results.json
• BOUNDARY_HAZARD_CATALOG.json