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Fermat's theorem: an interior differentiable local extremum has zero gradient
Statement
If is differentiable at an interior local maximum or minimum , then .
Facts & Assumptions
Given: A differentiable scalar field with a local extremum at .
The one-variable Fermat theorem gives derivative zero at an interior differentiable local extremum (Fermat's interior extremum theorem: if has a local extremum at a point interior to its domain and is differentiable at , then ).
The gradient consists of the coordinate partial derivatives (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case).
Proof
Restrict to each coordinate line through . The restriction has a local extremum at , so [L1] makes its derivative zero.
These derivatives are the entries of by [L2], so every entry vanishes.
Depends on
- Local and strict local extrema for scalar fields on Euclidean open sets
- Fermat's interior extremum theorem: if $f$ has a local extremum at a point $c$ interior to its domain and is differentiable at $c$, then $f'(c) = 0$
- The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case
Used by
- A strict subsolution can fail the supersolution lower-test condition Counterexample
- Minima of viscosity subsolutions need not be subsolutions Counterexample
- The Hamilton--Jacobi primitive of a Burgers solution Example
- The negative absolute value solves the eikonal equation in the viscosity sense Example
- Vanishing viscosity selects the Hopf--Lax solution for bounded data Example
- Exponential Lyapunov drift for the formal Langevin generator Lemma
- Langevin diffusion has a reversible density and compact minorization Lemma
- Radial gradient drift forces potential coercivity Lemma
- The Hessian is negative semidefinite at an interior local maximum Lemma
- Time-space barriers enforce the initial trace for the Cauchy problem Lemma
- ULA invariant laws have uniform exponential moments on the needed step ranges Lemma
- ULA's Gaussian kernel satisfies an exponential drift bound Lemma
- Classical solutions are viscosity solutions and differentiable viscosity solutions solve the equation pointwise Proposition
- An elementary polynomial sign-pattern bound Theorem
- Rates for randomized coordinate gradient descent and importance sampling Theorem
- Strong maximum principle for harmonic functions Theorem
- The score-matching integration-by-parts identity Theorem
Dependency tree · two levels
17 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Analysis, Convexity, and Optimization (standard reference, not scraped)