Optimization and Lagrange Multipliers: every key term you need (+ practice quiz)
25 flashcard terms for Multivariable Calculus Topic 5, written to match the course framework. Study them here, then drill them as interactive flashcards, or test yourself with the 15-question quiz — free, no account needed.
A point whose function value is at least as large as every value in some surrounding disk. The comparison is local, so a local maximum need not be the global one.
Local minimum of a multivariable function
A point whose function value is no greater than every value in some surrounding disk, again judged only against nearby competitors.
Critical point
An interior domain point where the gradient is the zero vector or where at least one partial derivative fails to exist. Every interior extremum must occur at one.
Stationary point
A critical point of the first kind, where all first partials exist and vanish simultaneously, so the tangent plane is horizontal there.
Saddle point
A critical point that is a maximum along one direction and a minimum along another, so it is neither a local maximum nor a local minimum despite a vanishing gradient.
Discriminant for the second derivative test
The quantity formed from the product of the two pure second partials minus the square of the mixed partial, evaluated at a critical point.
Second derivative test conclusion rules
A positive discriminant with a positive pure second partial gives a local minimum, a positive discriminant with a negative one gives a local maximum, and a negative discriminant gives a saddle.
Inconclusive second derivative test
Occurs when the discriminant is zero; the test gives no information and the point must be analysed directly with traces, inequalities or a change of variables.
Hessian matrix
The square matrix of all second partial derivatives. Its determinant is the discriminant in the two-variable case, and its eigenvalue signs classify critical points in general.
Absolute extremum on a region
The largest or smallest value attained anywhere on the region, which for a closed bounded region is guaranteed to exist by the extreme value theorem.
Closed region optimisation procedure
Evaluate the function at all interior critical points, then optimise it along every boundary piece, then compare all candidate values and pick the extremes.
Boundary parametrisation for optimisation
Reducing a boundary curve to a single parameter so the restricted function becomes one-variable, letting ordinary critical point analysis and endpoint checks apply.
Corner points of a boundary
Junctions between boundary pieces, where the restricted function may be nondifferentiable. They must be added to the candidate list explicitly.
Constrained optimisation
The problem of extremising a function subject to one or more equations linking the variables, so only points on the constraint set compete.
Constraint set
The level set of the constraint function, the only region where candidate points may lie. Its geometry, especially compactness, determines whether extremes must exist.
Method of Lagrange multipliers
At a constrained extremum the gradient of the objective is a scalar multiple of the gradient of the constraint, giving a system to solve together with the constraint equation.
Lagrange multiplier
The scalar linking the two gradients. Its value measures the sensitivity of the optimal objective value to a small relaxation of the constraint level.
Regularity condition for Lagrange multipliers
The constraint gradient must be nonzero at the candidate point; where it vanishes the method can miss genuine extrema and such points need separate checking.
Geometric reading of the multiplier condition
Parallel gradients mean the level set of the objective is tangent to the constraint curve, since crossing transversally would allow further improvement.
Two constraints in space
With two constraint surfaces the feasible set is their curve of intersection, and the objective gradient must be a linear combination of both constraint gradients.
Comparing Lagrange candidates
The method produces candidate points but never labels them; the objective must be evaluated at each and compared, with unbounded constraint sets checked for escape to infinity.
Unbounded constraint set warning
When the constraint set is not bounded, an extremum may fail to exist even though the multiplier system has solutions, so limiting behaviour must be examined.
Least squares fitting as an optimisation
Minimising a sum of squared residuals over the fitting parameters, a standard application where the gradient conditions produce a solvable linear system.
Extreme value theorem in several variables
A continuous function on a closed and bounded region attains both an absolute maximum and an absolute minimum on that region.
Nondifferentiable extremum
An extremum located where a partial derivative fails to exist, such as at a cone tip. Such points belong to the critical point list even though no gradient equation is solved.