Alphabeta MathMixed Partials, Taylor Formulae, and Extrema: Examples and Counterexamples
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1 · Prerequisites
Abstract Algebra
Linear Algebra
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Darboux, L'Hôpital, and Taylor's Theorem
- Foundations of the Real Numbers for Analysis
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Mixed Partials, Taylor Formulae, and Extrema
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Suprema and Infima
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Riemann Integral: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Counterexample
Peano's function has unequal mixed partials at the origin
Counterexample
Peano's surface has a strict minimum on every line through the origin but no local extremum
Counterexample
A smooth flat refinement has a strict minimum on every line through the origin but no local extremum
Counterexample
x²+y²(1-x)³ has a unique critical point, a strict local but nonglobal minimum
Example
The monkey saddle x³-3xy² has an indefinite higher-order critical point
Counterexample
A zero Hessian occurs at a strict minimum, a strict maximum, and a saddle
Example
A second-order Taylor polynomial computed from gradient and Hessian data
Example
A constrained extremum on an affine graph satisfies the graph Lagrange rule
Counterexample