Operations research, mathematical programming Proof

Proof of the Lagrange multiplier rule with one constraint

Builds on Lagrange multipliers Gradient Level set Linear independence C^k function characterised by continuous derivatives

Proof

We apply the theorem with several constraints to the single constraint $g - c$ .

Let $U$ , $n \ge 2$ , $f$ , $g$ , $c$ and $a$ be as in the statement. Define $G = g - c$ on $U$ . It is of class C^1, since $g$ is.
The level set $\set{x \in U : g(x) = c}$ is the set $M = \set{x \in U : G(x) = 0}$ .
The gradient of $G$ is $\nabla G = \nabla g$ , since $c$ is constant. So $\nabla G(a) \ne 0$ .
A single nonzero vector is linearly independent. Also $m = 1 < n$ , since $n \ge 2$ .
The restriction of $f$ to $M$ has a local extremum at $a$ . By the Lagrange multiplier theorem with $m = 1$ and the constraint $G$ , there is a unique $\lambda \in \R$ with $\nabla f(a) = \lambda \nabla G(a)$ .
By step 3, this says $\nabla f(a) = \lambda \nabla g(a)$ , and $\lambda$ is unique.

Idea

A level set $g = c$ is the zero set of $g - c$ , which has the same gradient as $g$ . One nonzero gradient is a linearly independent family, so the general theorem applies with one multiplier.

Reading path

To understand this page from the ground up, read these 481 pages in order. The foundations come first. The path takes the proofs Proof of the Steinitz exchange lemma by induction, Proof of the limit of a sum of sequences, Proof of Rolle's theorem, Proof of the derivative of a map into R^n by components, Proof of the approximation property of the supremum, Proof that every sequence has a monotone subsequence, by peaks, Proof of the decomposition of a permutation into transpositions, Proof of the implicit function theorem from the inverse function theorem, Proof that the symmetric group is generated by transpositions, Proof of the Laplace expansion along a column by transposition, Proof that open sets are the sets that are neighbourhoods of all their points, Proof that closed balls are closed, Proof of the diameter characterisation of Cauchy sequences, Proof of the commutativity of addition by induction, Proof of the Lebesgue number lemma by a finite cover of balls, Proof of the epsilon-delta form of continuity, Proof that equivalence classes form a partition, Proof of the triangle inequality for norms, Proof of the sum of a finite geometric series, Proof that a complex number times its conjugate is the square of its modulus, Proof of the Lagrange multiplier theorem by the implicit function theorem, Proof of the Cauchy–Schwarz inequality, Proof of the Heine–Borel theorem by sequential compactness, Proof that there is a natural number n with 1/n below any positive epsilon, Proof of the chain rule in several variables, Proof that the coordinate space F^n has dimension n, Proof that the determinant is multiplicative by expanding the rows, Proof that permuting rows multiplies the determinant by the sign, Proof that closed subsets of complete spaces are complete, Proof of the derivative as a little-o approximation, Proof that disjoint cycles commute, Proof that the derivative vanishes at a local extremum, Proof of the equality of complex numbers, Proof that a matrix is invertible iff its determinant is a unit, Proof that a linear map between normed spaces is continuous if and only if it is bounded, Proof of the determinant of a block triangular matrix by the Leibniz formula, Proof that a function is bijective iff it has a two-sided inverse, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of continuity via closed sets, Proof that a matrix with two equal rows has determinant zero, Proof of the double negation law by a truth table, Proof of the extreme value theorem by the Bolzano–Weierstrass theorem, Proof that compact subsets of a metric space are bounded, Proof of the Laplace expansion along a row by the Leibniz formula, Proof that compactness and sequential compactness agree in metric spaces, Proof that subsequences of a convergent sequence converge to the same limit, Proof of the uniqueness of limits, Proof that the coordinate map is a linear isomorphism, Proof of strong induction from ordinary induction, Proof that linear maps on finite-dimensional normed spaces are continuous, Proof that the determinant of a transpose equals the determinant, Proof of the laws of exponents in a group, Proof of the limit of a product of sequences, Proof of the inverse in terms of the adjugate, Proof that the absolute value is multiplicative, Proof of the squeeze theorem for sequences, Proof of the existence of square roots of nonnegative reals, Proof of the chain rule for a composition with a curve, Proof of the linearity of differentiation, Proof that a continuous image of a compact set is compact, Proof that compact subsets of a Hausdorff space are closed, Proof that the composition of continuous maps is continuous, Proof of epsilon-delta continuity at a point in metric spaces, Proof of the sequential characterization of limits, Proof of the description of the open sets of a metric space, Proof of the Banach fixed point theorem by iteration, Proof of the triangle inequality for real numbers, Proof that an intersection of subgroups is a subgroup, Proof of the epsilon-N definition of the limit of a sequence, Proof that compactness of a subset can be tested with open sets of the ambient space, Proof that the sign is a homomorphism, Proof of the existence of the infimum by negation, Proof of the equivalence of norms in finite dimension by compactness of the unit sphere, Proof of the determinant of the inverse, Proof that every spanning set contains a basis, by Zorn's lemma, Proof that the determinant of the identity matrix is one, Proof of Fermat's theorem on stationary points, Proof that limits preserve non-strict inequalities, Proof of the inverse function theorem by the contraction mapping principle, Proof that closed sets contain their limits, Proof of Zorn's lemma, Proof that metric spaces are Hausdorff, Proof that the derivative gives the directional derivatives, Proof that differentiable functions are continuous, Proof of the decomposition of a permutation into disjoint cycles, Proof of the product of a matrix and its adjugate by Laplace expansion, Proof of the sequence characterisation of uniform continuity, Proof of the epsilon-delta form of the limit of a function, Proof of the well-ordering principle from strong induction, Proof of the Bolzano–Weierstrass theorem by a monotone subsequence, Proof of the division algorithm, Proof of the Archimedean property in an Archimedean ordered field, Proof of the sign of a cycle, Proof that finite-dimensional normed spaces are complete, Proof that a continuous real function on a compact set attains its bounds, Proof of the Leibniz formula from the defining properties of the determinant, Proof of the uniqueness of inverses, Proof of the derivative of a map into a product, Proof that row rank equals column rank by a factorization A = CR, Proof that a decreasing sequence bounded below converges, Proof that the reals are Archimedean from the completeness axiom, Proof that convergent sequences are bounded, Proof of the monotone convergence theorem for sequences, Proof that continuity is equivalent to sequential continuity, Proof that a subspace of a finite-dimensional space has smaller or equal dimension, Proof that the determinant is multilinear in the rows, Proof that open balls are open, Proof of the boundedness theorem by Bolzano–Weierstrass, Proof that Lipschitz maps are uniformly continuous, Proof that C^k functions are characterised by continuous derivatives, Proof that continuous partial derivatives imply differentiability, Proof that a set is open if and only if its complement is closed, Proof of the subgroup criterion, Proof of the mean value theorem from Rolle's theorem, Proof of continuity via open sets in metric spaces, Proof of the mean value inequality, Proof of the constant multiple rule.

Affine combination concept
Basis concept
Binary operation definition
Column space concept
Complex numbers object
Complex numbers as pairs of reals concept
Construction of the reals from Cauchy sequences concept
Coordinate space object
Coordinate vector concept
Cycle notation concept
Definition by recursion concept
Dimension concept
Divisibility definition
Element definition
Euclidean metric example
Euclidean space object
Extended real number line object
Function concept
Function composition concept
Group concept
Identity matrix object
Imaginary unit object
Inner product concept
Inner product space concept
Integers object
Limit point of a set definition
Linear isomorphism concept
Linear map concept
Linear span concept
Local maximum definition
Mathematical induction concept
Matrix definition
Matrix multiplication concept
Multilinear map concept
Natural numbers object
Norm concept
Norm induced by an inner product concept
Ordered field definition
Peano axioms concept
Permutation concept
Permutation definition
Preimage of a set concept
Proof of Zorn's lemma proof
Proposition concept
Rank of a matrix concept
Rational numbers object
Real numbers object
Row space concept
Sequence definition
Set bounded above definition
Standard basis object
Transpose concept
Transposition concept
Vector space definition
p-metric on R^n example
Identity element definition
Real and imaginary parts definition
Convex set definition
Set definition
Fréchet derivative definition
Having a directional derivative definition
Total derivative concept
Function of several variables concept
Vector-valued function concept
Group homomorphism definition
Ring definition
Subgroup definition
Symmetric group definition
Ring of integers statement
Limit of a function definition
Local minimum definition
Block matrix definition
Matrix multiplication definition
Square matrix definition
Transpose definition
Successor function definition
Archimedean ordered field definition
Conjunction definition
Disjunction definition
Implication definition
Negation definition
Predicate definition
The reals form a field statement
Real sequence definition
Subsequence definition
Infimum definition
Supremum definition
Inner product definition
Linear combination definition
Linear map definition
Linear subspace definition
Norm definition
p-norm on R^n example
Inverse element definition
Complex conjugate definition
Empty set object
Family of sets definition
Ordered pair definition
Differentiable function of several variables concept
Local maximum and minimum definition
Local maximum and minimum concept
Partial derivative concept
Unit of a ring definition
Transposition definition
Limit of a function of several variables concept
Parametrised curve concept
Intersection definition
Union definition
Continuity at a point definition
Derivative definition
Set-builder notation concept
Proof that an intersection of subgroups is a subgroup proof
Invertible matrix definition
Addition of natural numbers definition
Biconditional definition
Truth table concept
Level set definition
Modus ponens concept
Existential quantifier definition
Universal quantifier definition
Bounded sequence definition
Cauchy sequence definition
Geometric sequence definition
Limit of a sequence definition
Monotone sequence definition
Proof of the Archimedean property in an Archimedean ordered field proof
The rationals form a field statement
The successor function is injective statement
Zero is not a successor statement
Inner product space definition
Linear independence definition
Composition of linear maps definition
Linear isomorphism definition
Linear span definition
Multilinear map definition
Normed vector space definition
Completeness axiom statement
Cartesian product definition
Gradient concept
Constrained optimisation concept
Jacobian matrix concept
Proof of the equality of complex numbers proof
Disjoint sets definition
Proof of the uniqueness of inverses proof
Partial derivative definition
Continuity in R^n definition
Differentiable function of several variables definition
Union of a family of sets definition
Continuous function definition
Axiom of extensionality statement
Differentiable function definition
Subset definition
Proof of the derivative as a little-o approximation proof
Proof of the commutativity of addition by induction proof
Mathematical proof concept
Uniqueness quantifier definition
Propositional logic concept
Archimedean property statement
Order on the natural numbers definition
Proof of the sum of a finite geometric series proof
Convergent sequence definition
Sequential continuity definition
The rationals form an ordered field statement
Column space definition
Row space definition
Norm induced by an inner product definition
Spanning set definition
Alternating multilinear map definition
Equality of complex numbers statement
Binary relation definition
Higher partial derivatives definition
Uniqueness of inverses statement
Jacobian matrix definition
Universal set definition
Differentiable on an open set definition
Derivative as a little-o approximation statement
Partition of a set definition
Topological space definition
Gradient definition
Proof of the derivative of a map into R^n by components proof
Order on the reals definition
Tautology definition
Commutativity of addition statement
Determinant definition
Sum of a finite geometric series statement
Proof of strong induction from ordinary induction proof
Finite-dimensional vector space definition
Basis definition
Proof of the Steinitz exchange lemma by induction proof
Antisymmetric relation definition
Reflexive relation definition
Symmetric relation definition
Transitive relation definition
Iterated derivative definition
Jacobian determinant definition
Sequential compactness definition
Function definition
Complement definition
Proof of the laws of exponents in a group proof
Proof of the subgroup criterion proof
Interval definition
Subspace topology definition
Proof that a matrix with two equal rows has determinant zero proof
Proof that the determinant of the identity matrix is one proof
Dimension definition
Logical equivalence definition
Coordinate vector definition
Dimension theorem for vector spaces statement
Strong induction statement
Minor definition
Proof that the determinant is multilinear in the rows proof
Steinitz exchange lemma statement
Equivalence relation definition
Preorder definition
Codomain definition
Function composition definition
Identity function definition
Injective function definition
Choice function definition
Image of a set definition
Preimage of a set definition
Restriction of a function definition
Subgroup criterion statement
Proof of the epsilon-delta form of the limit of a function proof
Proof of the epsilon-delta form of continuity proof
Determinant of a matrix with two equal rows statement
Laws of exponents in a group statement
Determinant of the identity matrix statement
Metric space definition
Determinant is multilinear in the rows statement
Cofactor definition
Rank of a matrix definition
Proof of the well-ordering principle from strong induction proof
Proof that the coordinate space F^n has dimension n proof
Proof that a subspace of a finite-dimensional space has smaller or equal dimension proof
Proof of the double negation law by a truth table proof
Partial order definition
Left inverse definition
Right inverse definition
Surjective function definition
Axiom of choice statement
Proof that disjoint cycles commute proof
Equivalence class definition
Epsilon-delta definition of a limit statement
Intersection of subgroups is a subgroup statement
Subspace metric definition
Epsilon-delta definition of continuity statement
Dimension of the coordinate space statement
Closed ball definition
Convergence of a sequence in a metric space definition
Lipschitz map definition
Open ball definition
Triangle inequality for a metric statement
Adjugate matrix definition
Diameter of a set definition
Double negation law statement
Well-ordering principle statement
Dimension of a subspace statement
Maximal element definition
Total order definition
Upper bound definition
Bijective function definition
Proof that equivalence classes form a partition proof
Disjoint cycles commute statement
Subgroup generated by a subset definition
Metric topology definition
Bounded set definition
Contraction mapping definition
Proof of the division algorithm proof
Proof by contradiction concept
Proof of the diameter characterisation of Cauchy sequences proof
Proof that row rank equals column rank by a factorization A = CR proof
Proof that every sequence has a monotone subsequence, by peaks proof
Chain definition
Proof that a function is bijective iff it has a two-sided inverse proof
Equivalence classes form a partition statement
Open set definition
Division algorithm statement
Epsilon characterisation of Cauchy sequences statement
Proof that the reals form an ordered field, with the Cauchy sequence construction proof
Row rank equals column rank statement
Every sequence has a monotone subsequence statement
Bijective iff it has a two-sided inverse statement
Hausdorff space definition
Open cover definition
Closed set definition
Continuous map of topological spaces definition
Proof that open sets are the sets that are neighbourhoods of all their points proof
Proof of the decomposition of a permutation into disjoint cycles proof
Proof that the derivative gives the directional derivatives proof
Equivalent metrics definition
Zorn's lemma statement
Proof of the description of the open sets of a metric space proof
Complete metric space definition
The reals form an ordered field statement
Compact space definition
Proof that the composition of continuous maps is continuous proof
Proof that a set is open if and only if its complement is closed proof
Open sets characterised by neighbourhoods statement
Derivative gives the directional derivatives statement
Equivalent norms definition
Decomposition of a permutation into disjoint cycles statement
Open sets of a metric space statement
Proof that the coordinate map is a linear isomorphism proof
Absolute value definition
Proof that every spanning set contains a basis, by Zorn's lemma proof
Proof of the approximation property of the supremum proof
Proof of the existence of the infimum by negation proof
Proof of the existence of square roots of nonnegative reals proof
Complement of an open set is closed statement
Composition of continuous maps is continuous statement
Continuous linear map definition
Compact set definition
Continuity at a point of a topological space definition
Banach space definition
Proof of the decomposition of a permutation into transpositions proof
Proof of the triangle inequality for real numbers proof
Proof that the absolute value is multiplicative proof
Proof that closed sets contain their limits proof
Proof that closed balls are closed proof
Proof that open balls are open proof
Every spanning set contains a basis statement
Proof of the epsilon-N definition of the limit of a sequence proof
Isomorphism with the coordinate space statement
Existence of square roots of nonnegative reals statement
Approximation property of the supremum statement
Existence of the infimum statement
Proof that compactness of a subset can be tested with open sets of the ambient space proof
Proof of continuity via closed sets proof
Decomposition of a permutation into transpositions statement
Closed sets contain their limits statement
Triangle inequality statement
Absolute value is multiplicative statement
Closed balls are closed statement
Open balls are open statement
Square root definition
Epsilon-N definition of the limit of a sequence statement
Proof that the reals are Archimedean from the completeness axiom proof
Sign of a permutation definition
Continuity via closed sets statement
Compactness via open covers statement
Modulus of a complex number definition
Proof of the limit of a sum of sequences proof
Proof that closed subsets of complete spaces are complete proof
Proof of the uniqueness of limits proof
Proof that the symmetric group is generated by transpositions proof
Proof of epsilon-delta continuity at a point in metric spaces proof
Proof of the squeeze theorem for sequences proof
Proof of the Lebesgue number lemma by a finite cover of balls proof
Proof of continuity via open sets in metric spaces proof
Proof that limits preserve non-strict inequalities proof
Proof that differentiable functions are continuous proof
Proof that metric spaces are Hausdorff proof
Proof that convergent sequences are bounded proof
The reals are Archimedean statement
Proof that subsequences of a convergent sequence converge to the same limit proof
Proof of the monotone convergence theorem for sequences proof
Proof that compact subsets of a Hausdorff space are closed proof
Proof that a complex number times its conjugate is the square of its modulus proof
Proof that a continuous image of a compact set is compact proof
The symmetric group is generated by transpositions statement
Proof that the sign is a homomorphism proof
Closed subsets of complete spaces are complete statement
Lebesgue number lemma statement
Proof of the sign of a cycle proof
Epsilon-delta continuity in metric spaces statement
Proof that compact subsets of a metric space are bounded proof
Metric spaces are Hausdorff statement
Proof that compactness and sequential compactness agree in metric spaces proof
Differentiable functions are continuous statement
Limit of a sum of sequences statement
Proof of the sequence characterisation of uniform continuity proof
Uniqueness of limits statement
Limits preserve non-strict inequalities statement
Convergent sequences are bounded statement
Proof of the triangle inequality for norms proof
Squeeze theorem statement
Subsequences of a convergent sequence converge to the same limit statement
Proof that there is a natural number n with 1/n below any positive epsilon proof
Monotone convergence theorem for sequences statement
Compact subsets of a Hausdorff space are closed statement
A complex number times its conjugate statement
Continuous image of a compact set is compact statement
Sign of a cycle statement
Sign is a homomorphism statement
Compactness and sequential compactness agree in metric spaces statement
Continuity via open sets statement
Compact subsets of a metric space are bounded statement
Proof of the limit of a product of sequences proof
Epsilon-delta uniform continuity statement
Proof that permuting rows multiplies the determinant by the sign proof
There is a natural number n with 1/n below any positive epsilon statement
Proof that a decreasing sequence bounded below converges proof
The sequence r^n tends to zero for |r| < 1 example
Triangle inequality for norms statement
Proof that a linear map between normed spaces is continuous if and only if it is bounded proof
Proof of the Banach fixed point theorem by iteration proof
Permuting rows multiplies the determinant by the sign statement
Proof that Lipschitz maps are uniformly continuous proof
Proof of the sequential characterization of limits proof
Proof that continuity is equivalent to sequential continuity proof
Decreasing sequence bounded below converges statement
Proof of the Cauchy–Schwarz inequality proof
Limit of a product of sequences statement
Banach fixed point theorem statement
Lipschitz maps are uniformly continuous statement
Continuity is equivalent to sequential continuity statement
Sequential characterization of limits statement
Proof of the Leibniz formula from the defining properties of the determinant proof
Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
Cauchy–Schwarz inequality statement
Bounded linear maps are continuous statement
Leibniz formula for determinants statement
Proof of the linearity of differentiation proof
Bolzano–Weierstrass theorem statement
Proof of the derivative of a map into a product proof
Proof of the determinant of a block triangular matrix by the Leibniz formula proof
Proof of the Heine–Borel theorem by sequential compactness proof
Linearity of differentiation statement
Operator norm definition
Proof of the Laplace expansion along a row by the Leibniz formula proof
Proof that the determinant is multiplicative by expanding the rows proof
Proof that the determinant of a transpose equals the determinant proof
Proof of the boundedness theorem by Bolzano–Weierstrass proof
Completeness of the real numbers statement
Derivative of a map into a product statement
Proof of the constant multiple rule proof
Heine–Borel theorem statement
Boundedness theorem statement
Determinant of a block triangular matrix statement
Determinant is multiplicative statement
Laplace expansion along a row statement
Determinant of a transpose statement
Constant multiple rule statement
Derivative of a map into R^n by components statement
Proof of the determinant of the inverse proof
Proof of the Laplace expansion along a column by transposition proof
Proof of the extreme value theorem by the Bolzano–Weierstrass theorem proof
Proof that a continuous real function on a compact set attains its bounds proof
Extreme value theorem statement
Continuous real function on a compact set attains its bounds statement
Determinant of the inverse statement
Laplace expansion along a column statement
Proof of Fermat's theorem on stationary points proof
Proof of the equivalence of norms in finite dimension by compactness of the unit sphere proof
Proof of the product of a matrix and its adjugate by Laplace expansion proof
Fermat's theorem on stationary points statement
Proof that the derivative vanishes at a local extremum proof
Equivalence of norms in finite dimension statement
Product of a matrix and its adjugate statement
Proof of Rolle's theorem proof
Rolle's theorem statement
Derivative vanishes at a local extremum statement
Proof that linear maps on finite-dimensional normed spaces are continuous proof
Proof that finite-dimensional normed spaces are complete proof
Proof of the inverse in terms of the adjugate proof
Linear maps on finite-dimensional spaces are continuous statement
Finite-dimensional normed spaces are complete statement
Proof of the mean value theorem from Rolle's theorem proof
Inverse in terms of the adjugate statement
Mean value theorem statement
Proof that a matrix is invertible iff its determinant is a unit proof
Proof of the chain rule in several variables proof
Proof that continuous partial derivatives imply differentiability proof
Chain rule in several variables statement
A matrix is invertible iff its determinant is a unit statement
Derivative of the inverse function in several variables statement
Continuous partial derivatives imply differentiability statement
Proof of the chain rule for a composition with a curve proof
Chain rule for a composition with a curve statement
Proof that C^k functions are characterised by continuous derivatives proof
C^k function characterised by continuous derivatives statement
Proof of the mean value inequality proof
Mean value inequality statement
Proof of the inverse function theorem by the contraction mapping principle proof
Inverse function theorem statement
Proof of the implicit function theorem from the inverse function theorem proof
Implicit function theorem statement
Proof of the Lagrange multiplier theorem by the implicit function theorem proof
Lagrange multipliers statement
Proof of the Lagrange multiplier rule with one constraint proof

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Lagrange multiplier with one constraint

Tag	`1GSGEE9PNE`
URI	`m2://1GSGEE9PNE`
Kind	proof
Writing	W0 stub
MSC	90C46
Proves	Lagrange multiplier with one constraint
Uses	Lagrange multipliers, Gradient, Level set, Linear independence, C^k function characterised by continuous derivatives
Changed	4 November 2026

Proof of the Lagrange multiplier rule with one constraint

Proof

Idea

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Where this leads

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