PHYS2150 Chap.3 Extrema, Constraints and Error Estimation
Extrema, Constraints and Error Estimation
One workflow, three decisions
This chapter separates candidate generation, classification and comparison. Derivative equations generate locations, curvature tests classify local behaviour, and domain-wide value checks establish global extrema.
Error estimation then uses the same derivatives for sensitivity rather than optimisation, so the question being answered must remain explicit.
Candidates are not conclusions
Optimisation starts by reading the domain. At a differentiable interior extremum, the first partial derivatives vanish, but the reverse statement is false: a stationary point can be a maximum, minimum or saddle.
Points where derivatives fail to exist may also require attention. The practical method is therefore to generate a complete candidate set before classifying anything. Record interior stationary points, boundary candidates, corners, endpoints and singular points separately.
A global conclusion follows only after the objective has been evaluated at every admissible candidate.
The Hessian classifies local curvature
For a two-variable function, the second-derivative discriminant compares the product of the pure second derivatives with the square of the mixed derivative.
A positive discriminant indicates consistent bending; the sign of the x second derivative then separates a local minimum from a local maximum. A negative discriminant indicates a saddle because the surface bends in opposite senses along different directions. A zero discriminant is inconclusive rather than a new classification.
In that case, inspect the function along suitable curves or use higher-order terms instead of forcing the test to decide.
Boundaries complete the global search
A closed bounded region can attain an extremum away from an interior stationary point. Each smooth boundary segment becomes a one-variable optimisation problem after a parametrisation or substitution.
Corners and endpoints must be checked directly because an ordinary derivative condition need not hold there. Keep the original restrictions visible while solving. A candidate that violates positivity, geometry or a stated interval is not rescued by satisfying the stationary equations.
The completed candidate ledger should show both the coordinates and the corresponding objective value, making the final comparison explicit.
Lagrange multipliers preserve the constraint
For a smooth equality constraint, the objective and constraint gradients are parallel at a regular constrained extremum.
Solve the parallel-gradient equations together with the original constraint; omitting the constraint produces points that may not be feasible. The constraint gradient must also be nonzero at the candidate for the regular method to apply. Geometrically, an objective level curve touches the constraint curve with a common tangent.
The multiplier can be interpreted as local sensitivity of the optimal value to the constraint level when the units and regularity assumptions support that reading.
Differentials turn tolerances into output error
Error estimation reuses local derivatives to measure sensitivity.
Differentiate the physical model symbolically, insert central measurements and then combine derivative contributions with their input tolerances. A worst-case first-order bound adds absolute contributions, while a root-sum-square combination needs an explicit independence assumption. Absolute uncertainty keeps the output unit; relative and percentage uncertainty are dimensionless.
Rank the separate contributions before reporting the total, because the dominant term identifies which measurement improvement would most reduce uncertainty. Finish by matching significant figures to the estimated precision rather than displaying unsupported calculator digits.
What this chapter covers
- 01
Critical points and Hessian tests
- 02
Closed-region boundary checks
- 03
Lagrange multiplier equations
- 04
Differentials and sensitivity
- 05
Absolute and relative uncertainty
- 06
Constraint regularity and sensitivity
Worked example · free
Optimise on a constraint
- 2Write both gradients.
- 4Solve with the constraint.
- 2Compare objective values.
Key terms
- Hessian
- The matrix of second partial derivatives describing local curvature.
- Lagrange Multiplier
- A scalar enforcing parallel objective and constraint gradients at a constrained candidate.
- Critical Point
- A domain point where both first partial derivatives vanish or at least one does not exist.
- Saddle Point
- A stationary point that is neither a local maximum nor a local minimum.
- Total Differential
- The first-order combination of input changes used to approximate an output change.
- Relative Error
- An absolute error divided by the magnitude of the corresponding measured or calculated quantity.
Extrema, Constraints and Error Estimation FAQ
Why is solving the gradient equation insufficient?
A zero gradient addresses only differentiable interior points and generates candidates rather than conclusions. A closed region can attain global extrema on smooth boundaries, corners, endpoints or singular points, so those locations must be analysed separately and their objective values compared.
What does a zero Hessian discriminant mean?
The standard second-derivative test is inconclusive. It does not mean the point is flat, a saddle or not an extremum. Examine the function directly along suitable directions, compare signs of nearby changes, or use higher-order terms that the discriminant ignores.
How is an uncertainty estimate checked?
Each derivative contribution must reduce to the output unit. A first-order worst-case bound should scale linearly when every tolerance is scaled, and the reported precision should reflect the resulting uncertainty rather than calculator digits. Rank contributions to identify the measurement controlling the total error.
When can the Lagrange multiplier equations miss a point?
The regular method assumes the constraint gradient is nonzero at the candidate. If that gradient vanishes, parallel-gradient equations may not generate every constrained extremum. Inspect the point separately, keep the original constraint active, and compare its objective value with all other admissible candidates.
Exam move
Use a candidate ledger for every optimisation problem. Write the domain first, then create separate rows for differentiable interior points, nonsmooth points, each boundary segment, endpoints and corners. Beside every candidate, record the restriction check and objective value.
For a smooth constraint, keep the original constraint beside the multiplier equations and test that its gradient is nonzero before relying on the regular Lagrange condition. Practise explaining the geometry of parallel gradients instead of treating lambda as an unexplained extra variable.
For uncertainty questions, differentiate before inserting numbers, annotate every quantity with units, and calculate each sensitivity contribution separately. Decide whether the problem supports a worst-case sum or a statistical combination rather than switching formulas by habit.
Rank contributions, check that scaling all tolerances scales the first-order estimate, and round the final result to a precision justified by the uncertainty. Revisit any problem where a candidate was discarded without a stated domain reason or where a local classification was mistaken for a global comparison.
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