Calculus of variations and optimal control; optimization Proof

Proof of the second derivative test in several variables

Builds on Second-order Taylor expansion Hessian matrix Schwarz's theorem for C^2 functions Definite and indefinite quadratic forms Local maximum and minimum Saddle point Eigenvector Heine–Borel theorem Continuous real function on a compact set attains its bounds C^k function characterised by continuous derivatives

Proof

We compare $f$ near $a$ with its second-order Taylor polynomial.

Let $U$ , $f$ , $a$ and the Hessian matrix $H = H_f(a)$ be as in the statement. Since $f$ is of class C^2, Schwarz's theorem shows that $H$ is symmetric. Its quadratic form $Q(h) = h^{\mathsf{T}} H h$ is as in the definition of definite forms.
Since $\nabla f(a) = 0$ , the second-order Taylor expansion gives $f(a + h) - f(a) = \tfrac{1}{2} Q(h) + R(h)$ for $a + h \in U$ , with $R(h) / \norm{h}^2 \to 0$ as $h \to 0$ .
Part 1. Assume that $H$ is positive definite. Let $S = \set{h \in \R^n : \norm{h} = 1}$ be the unit sphere. It is closed and bounded, so it is compact by the Heine–Borel theorem.
The function $Q$ is a polynomial in the coordinates of $h$ , so it is continuous. By the extreme value theorem, there is $h_0 \in S$ with $Q(h) \ge Q(h_0)$ for all $h \in S$ . Let $m = Q(h_0)$ . Then $m > 0$ , since $h_0 \ne 0$ and $Q$ is positive definite.
For $h \ne 0$ , the vector $h / \norm{h}$ lies in $S$ , and $Q(h) = \norm{h}^2 Q(h / \norm{h})$ . So $Q(h) \ge m \norm{h}^2$ for every $h \in \R^n$ .
By step 2, there is $\delta > 0$ such that $a + h \in U$ and $\abs{R(h)} < \tfrac{m}{4} \norm{h}^2$ whenever $0 < \norm{h} < \delta$ .
For such $h$ , steps 2, 5 and 6 give $f(a + h) - f(a) \ge \tfrac{m}{2} \norm{h}^2 - \tfrac{m}{4} \norm{h}^2 = \tfrac{m}{4} \norm{h}^2 > 0$ .
So $f(x) > f(a)$ for all $x \in U$ with $0 < \norm{x - a} < \delta$ . This means that $f$ has a strict local minimum at $a$ .
Part 2. Assume that $H$ is negative definite. The function $-f$ is of class $C^2$ , has gradient $0$ at $a$ , and has Hessian $-H$ , which is positive definite.
By Part 1, $-f$ has a strict local minimum at $a$ . So $f$ has a strict local maximum at $a$ .
Part 3. Let $\lambda > 0$ and $\mu < 0$ be eigenvalues of $H$ . The real matrices $H - \lambda I$ and $H - \mu I$ are singular. So there are real eigenvectors $v, w \in \R^n$ with $H v = \lambda v$ and $H w = \mu w$ . Then $Q(v) = \lambda \norm{v}^2 > 0$ and $Q(w) = \mu \norm{w}^2 < 0$ .
For real $t \ne 0$ small enough that $a + t v \in U$ , step 2 gives $\frac{f(a + t v) - f(a)}{t^2 \norm{v}^2} = \frac{\lambda}{2} + \frac{R(t v)}{\norm{t v}^2}$ .
As $t \to 0$ , the right side tends to $\lambda / 2 > 0$ . So there is $\eta > 0$ with $f(a + t v) > f(a)$ for $0 < \abs{t} < \eta$ .
In the same way, using $w$ and $\mu < 0$ , there is $\eta' > 0$ with $f(a + t w) < f(a)$ for $0 < \abs{t} < \eta'$ .
Let $\varepsilon > 0$ . By steps 13 and 14, the ball of radius $\varepsilon$ around $a$ contains a point $x \in U$ with $f(x) > f(a)$ . It also contains a point $y \in U$ with $f(y) < f(a)$ .
So $a$ is neither a local maximum nor a local minimum of $f$ . Since $\nabla f(a) = 0$ , the point $a$ is a saddle point of $f$ .

Idea

At a critical point the linear term of the Taylor expansion is zero, so $f(a + h) - f(a)$ is $\tfrac{1}{2} h^{\mathsf{T}} H h$ plus a remainder that is small compared to $\norm{h}^2$ . A positive definite form is at least $m \norm{h}^2$ , by compactness of the sphere, so it wins against the remainder. For an indefinite Hessian, moving along eigenvectors makes $f$ go up in one direction and down in another.

Reading path

To understand this page from the ground up, read these 409 pages in order. The foundations come first. The path takes the proofs Proof that Darboux and Riemann integrals agree, Proof of integration by parts, 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 that open sets are the sets that are neighbourhoods of all their points, 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 of the power rule by induction, Proof of the triangle inequality for norms, Proof that a complex number times its conjugate is the square of its modulus, 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 of the chain rule with an auxiliary difference quotient, Proof of the derivative as a little-o approximation, Proof of the second-order Taylor expansion, Proof of the equality of complex numbers, Proof that continuous functions on a closed bounded interval are uniformly continuous, Proof that a linear map between normed spaces is continuous if and only if it is bounded, 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 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 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 of the limit of a product of sequences, Proof of the linearity of the derivative, Proof that the absolute value is multiplicative, Proof of the squeeze theorem for sequences, Proof of the derivative of a continuous linear map, 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 of the mean value theorem for integrals, 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 Riemann's criterion for integrability, Proof of the sequential characterization of limits, Proof of Taylor's theorem with the integral remainder by the fundamental theorem of calculus, Proof of the description of the open sets of a metric space, Proof of the triangle inequality for real numbers, Proof of the epsilon-N definition of the limit of a sequence, Proof of the first fundamental theorem of calculus, Proof of Schwarz's theorem for C^2 functions by the mean value theorem, Proof that compactness of a subset can be tested with open sets of the ambient space, 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 that every spanning set contains a basis, by Zorn's lemma, Proof of Fermat's theorem on stationary points, Proof that limits preserve non-strict inequalities, Proof that closed sets contain their limits, Proof of the linearity of the integral, Proof of Zorn's lemma, Proof that metric spaces are Hausdorff, Proof that the derivative gives the directional derivatives, Proof of the second fundamental theorem of calculus, Proof of the intermediate value theorem by the supremum, Proof that differentiable functions are continuous, Proof of Taylor's theorem in several variables by restriction to a line, 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 Archimedean property in an Archimedean ordered field, Proof that a continuous real function on a compact set attains its bounds, Proof of the derivative of a map into a product, Proof that a composition of continuous functions is continuous, Proof of the additivity of the integral over intervals, 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 product rule, Proof of the monotone convergence theorem for sequences, Proof that continuity is equivalent to sequential continuity, Proof of the symmetry of second derivatives by a second difference, Proof that open balls are open, Proof of the monotonicity of the integral, 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 fundamental theorem of calculus, Proof of the mean value theorem from Rolle's theorem, Proof that sums and products of continuous functions are continuous, Proof of continuity via open sets in metric spaces, Proof that continuous functions are Riemann integrable, Proof of the mean value inequality, Proof of the constant multiple rule.

Basis 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
Definite and indefinite quadratic forms concept
Definition by recursion concept
Element definition
Euclidean metric example
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Extended real number line object
Function concept
Function composition concept
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Inner product concept
Inner product space concept
Integers object
Limit point of a set definition
Linear isomorphism concept
Linear map concept
Local maximum definition
Mathematical induction concept
Multinomial coefficient definition
Natural numbers object
Norm concept
Norm induced by an inner product concept
Ordered field definition
Partition of an interval definition
Peano axioms concept
Proof of Zorn's lemma proof
Proposition concept
Rational numbers object
Real numbers object
Sequence definition
Set bounded above definition
Vector space definition
p-metric on R^n example
Real and imaginary parts definition
Set definition
Fréchet derivative definition
Having a directional derivative definition
Total derivative concept
Function of several variables concept
Vector-valued function concept
Ring of integers statement
Limit of a function definition
Local minimum definition
Successor function definition
Archimedean ordered field definition
Mesh of a partition definition
Refinement of a partition definition
Riemann sum 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
Complex conjugate definition
Empty set object
Family of sets definition
Ordered pair definition
Differentiable function of several variables concept
Local maximum and minimum definition
Partial derivative concept
Limit of a function of several variables concept
Continuity at a point definition
Derivative definition
Intersection definition
One-sided limit definition
Parametrised curve concept
Union definition
Riemann integrable function definition
Addition of natural numbers definition
Proof of the derivative of a continuous linear map proof
Biconditional definition
Truth table concept
Modus ponens concept
Existential quantifier definition
Proof of the Archimedean property in an Archimedean ordered field proof
Universal quantifier definition
The rationals form a field statement
The successor function is injective statement
Zero is not a successor statement
Bounded sequence definition
Limit of a sequence definition
Monotone sequence definition
Inner product space definition
Linear independence definition
Composition of linear maps definition
Endomorphism definition
Linear isomorphism definition
Linear span definition
Multilinear map definition
Normed vector space definition
Completeness axiom statement
Cartesian product definition
Proof of the equality of complex numbers proof
Gradient concept
Jacobian matrix concept
Partial derivative definition
Continuous function definition
Continuity in R^n definition
Differentiable function of several variables definition
Antiderivative definition
Derivative of a continuous linear map statement
Differentiable function definition
One-sided derivative definition
Union of a family of sets definition
Subset definition
Riemann integral definition
Proof of the commutativity of addition by induction proof
Archimedean property statement
Mathematical proof concept
Proof of the derivative as a little-o approximation proof
Propositional logic concept
Order on the natural numbers definition
Uniqueness quantifier definition
The rationals form an ordered field statement
Convergent sequence definition
Sequential continuity definition
Upper and lower Darboux sums definition
Norm induced by an inner product definition
Eigenvalue definition
Spanning set definition
Equality of complex numbers statement
Higher partial derivatives definition
Binary relation definition
Proof that a composition of continuous functions is continuous proof
Differentiable on an open set definition
Universal set definition
Gradient definition
Proof of the derivative of a map into R^n by components proof
Topological space definition
Derivative function definition
Derivative as a little-o approximation statement
Order on the reals definition
Definite integral definition
Commutativity of addition statement
Tautology definition
Proof of strong induction from ordinary induction proof
Upper and lower Darboux integrals definition
Eigenvector definition
Finite-dimensional vector space definition
Basis definition
Hessian matrix definition
Multi-index notation concept
Antisymmetric relation definition
Reflexive relation definition
Transitive relation definition
Iterated derivative definition
Saddle point definition
Sequential compactness definition
Complement definition
Higher derivative definition
Subspace topology definition
Interval definition
Function definition
Logical equivalence definition
Coordinate vector definition
Strong induction statement
Preorder definition
Continuously differentiable function definition
Taylor polynomial definition
Proof of the epsilon-delta form of continuity proof
Proof of the epsilon-delta form of the limit of a function proof
Choice function definition
Codomain definition
Function composition definition
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Injective function definition
Image of a set definition
Preimage of a set definition
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Proof of the well-ordering principle from strong induction proof
Proof of the double negation law by a truth table proof
Partial order definition
Epsilon-delta definition of continuity statement
Epsilon-delta definition of a limit statement
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Right inverse definition
Axiom of choice statement
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Closed ball definition
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Triangle inequality for a metric statement
Double negation law statement
Diameter of a set definition
Well-ordering principle statement
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Upper bound definition
Uniform continuity definition
Composition of continuous functions is continuous statement
Bijective function definition
Bounded set definition
Metric topology definition
Proof by contradiction concept
Proof that every sequence has a monotone subsequence, by peaks proof
Chain definition
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Open set definition
Proof that a function is bijective iff it has a two-sided inverse proof
Proof that the reals form an ordered field, with the Cauchy sequence construction proof
Every sequence has a monotone subsequence statement
Bijective iff it has a two-sided inverse statement
Hausdorff space definition
Proof that the derivative gives the directional derivatives proof
Closed set definition
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Equivalent metrics definition
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Proof that open sets are the sets that are neighbourhoods of all their points proof
Zorn's lemma statement
Proof of the description of the open sets of a metric space proof
The reals form an ordered field statement
Derivative gives the directional derivatives statement
Proof that a set is open if and only if its complement is closed proof
Compact space definition
Equivalent norms definition
Proof that the composition of continuous maps is continuous proof
Open sets characterised by neighbourhoods 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
Continuous linear map definition
Composition of continuous maps is continuous statement
Compact set definition
Proof that closed sets contain their limits proof
Continuity at a point of a topological space definition
Proof of the triangle inequality for real numbers proof
Proof that the absolute value is multiplicative proof
Proof that open balls are open proof
Proof of the epsilon-N definition of the limit of a sequence proof
Isomorphism with the coordinate space statement
Every spanning set contains a basis statement
Approximation property of the supremum statement
Existence of the infimum statement
Existence of square roots of nonnegative reals statement
Proof that compactness of a subset can be tested with open sets of the ambient space proof
Closed sets contain their limits statement
Proof of continuity via closed sets proof
Triangle inequality statement
Absolute value is multiplicative statement
Open balls are open statement
Square root definition
Epsilon-N definition of the limit of a sequence statement
Proof of the intermediate value theorem by the supremum proof
Proof that the reals are Archimedean from the completeness axiom proof
Modulus of a complex number definition
Continuity via closed sets statement
Compactness via open covers statement
Proof of the limit of a sum of sequences proof
Proof that differentiable functions are continuous proof
Proof of the uniqueness of limits proof
Intermediate value theorem statement
Proof of the squeeze theorem for sequences proof
Proof of continuity via open sets in metric spaces proof
Proof that limits preserve non-strict inequalities proof
Proof of epsilon-delta continuity at a point in metric spaces proof
Proof of the Lebesgue number lemma by a finite cover of balls proof
Proof that metric spaces are Hausdorff proof
Proof of the linearity of the integral 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 a complex number times its conjugate is the square of its modulus proof
Proof that compact subsets of a Hausdorff space are closed proof
Proof that a continuous image of a compact set is compact proof
Proof that compact subsets of a metric space are bounded proof
Differentiable functions are continuous statement
Epsilon-delta continuity in metric spaces statement
Lebesgue number lemma statement
Metric spaces are Hausdorff statement
Proof that compactness and sequential compactness agree in metric spaces proof
Proof of the sequence characterisation of uniform continuity proof
Limit of a sum of sequences statement
Proof of the triangle inequality for norms proof
Linearity of the integral statement
Uniqueness of limits statement
Proof of the monotonicity of the integral proof
Limits preserve non-strict inequalities statement
Squeeze theorem statement
Proof that there is a natural number n with 1/n below any positive epsilon proof
Convergent sequences are bounded statement
Proof that Darboux and Riemann integrals agree proof
Subsequences of a convergent sequence converge to the same limit statement
Monotone convergence theorem for sequences statement
A complex number times its conjugate statement
Compact subsets of a Hausdorff space are closed statement
Continuous image of a compact set is compact statement
Compact subsets of a metric space are bounded statement
Continuity via open sets statement
Compactness and sequential compactness agree in metric spaces statement
Proof of the limit of a product of sequences proof
Epsilon-delta uniform continuity statement
Monotonicity of the integral statement
Darboux and Riemann integrals agree statement
There is a natural number n with 1/n below any positive epsilon statement
Triangle inequality for norms statement
Proof that a decreasing sequence bounded below converges proof
Proof that a linear map between normed spaces is continuous if and only if it is bounded proof
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
Proof of the Cauchy–Schwarz inequality proof
Decreasing sequence bounded below converges statement
Limit of a product of sequences statement
Proof of Riemann's criterion for integrability proof
Lipschitz maps are uniformly continuous statement
Continuity is equivalent to sequential continuity statement
Sequential characterization of limits statement
Riemann's criterion for integrability statement
Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
Cauchy–Schwarz inequality statement
Bounded linear maps are continuous statement
Proof of the chain rule with an auxiliary difference quotient proof
Proof of the product rule proof
Proof of the linearity of differentiation proof
Proof that sums and products of continuous functions are continuous proof
Bolzano–Weierstrass theorem statement
Proof of the additivity of the integral over intervals proof
Product of continuous functions is continuous statement
Proof of the derivative of a map into a product proof
Chain rule statement
Product rule statement
Proof of the linearity of the derivative proof
Linearity of differentiation statement
Proof that continuous functions on a closed bounded interval are uniformly continuous proof
Operator norm definition
Proof of the Heine–Borel theorem by sequential compactness proof
Additivity of the integral over intervals statement
Proof of the boundedness theorem by Bolzano–Weierstrass proof
Derivative of a map into a product statement
Linearity of the derivative statement
Continuous functions on a closed bounded interval are uniformly continuous statement
Proof of the constant multiple rule proof
Heine–Borel theorem statement
Proof of the power rule by induction proof
Boundedness theorem statement
Proof of the first fundamental theorem of calculus proof
Constant multiple rule statement
Derivative of a map into R^n by components statement
Power rule statement
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
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 that continuous functions are Riemann integrable proof
Fermat's theorem on stationary points statement
Continuous functions are Riemann integrable statement
Proof of Rolle's theorem proof
Equivalence of norms in finite dimension statement
Rolle's theorem statement
Proof of the mean value theorem for integrals proof
Proof that linear maps on finite-dimensional normed spaces are continuous proof
Linear maps on finite-dimensional spaces are continuous statement
Proof of the mean value theorem from Rolle's theorem proof
Mean value theorem for integrals statement
Mean value theorem statement
First fundamental theorem of calculus statement
Proof of the chain rule in several variables proof
Proof that continuous partial derivatives imply differentiability proof
Chain rule in several variables statement
Proof of the second fundamental theorem of calculus proof
Continuous partial derivatives imply differentiability statement
Proof of the chain rule for a composition with a curve proof
Second fundamental theorem of calculus statement
Proof of the fundamental theorem of calculus proof
Chain rule for a composition with a curve statement
Proof of integration by parts proof
Proof that C^k functions are characterised by continuous derivatives proof
Integration by parts statement
Fundamental theorem of calculus statement
C^k function characterised by continuous derivatives statement
Proof of the mean value inequality proof
Proof of Schwarz's theorem for C^2 functions by the mean value theorem proof
Proof of Taylor's theorem with the integral remainder by the fundamental theorem of calculus proof
Mean value inequality statement
Taylor's theorem with the integral remainder statement
Proof of the symmetry of second derivatives by a second difference proof
Symmetry of second derivatives statement
Schwarz's theorem for C^2 functions statement
Proof of Taylor's theorem in several variables by restriction to a line proof
Taylor's theorem in several variables statement
Proof of the second-order Taylor expansion proof
Second-order Taylor expansion statement
Proof of the second derivative test in several variables proof

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Second derivative test in several variables

Tag	`T2YZ93EKXE`
URI	`m2://T2YZ93EKXE`
Kind	proof
Writing	W0 stub
MSC	49K10
Proves	Second derivative test in several variables
Uses	Second-order Taylor expansion, Hessian matrix, Schwarz's theorem for C^2 functions, Definite and indefinite quadratic forms, Local maximum and minimum, Saddle point, Eigenvector, Heine–Borel theorem, Continuous real function on a compact set attains its bounds, C^k function characterised by continuous derivatives
Changed	4 November 2026

Proof of the second derivative test in several variables

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