Local minimum
Builds on Local maximum
Definition
Let , let , and let . The function has a local minimum at if there is with for all with . It is a strict local minimum if for all such .
Remarks
So has a local minimum at exactly when has a local maximum at .
For example, has a strict local minimum at .
Mathlib defines it as IsLocalMin.
Reading path
To understand this page from the ground up, read these 2 pages in order. The foundations come first.
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- no evidence level
- Local maximum definition
- Local minimum definition
Where this leads
117 pages build on this one.
- Power series are differentiable inside the radius of convergence
- A function with zero derivative is constant
- Natural logarithm
- A differentiable function is convex iff its derivative is increasing
- Taylor's theorem with the Cauchy remainder
- Euler–Mascheroni constant
- The logarithm grows slower than any power
- Derivative of sine
- 1 + x is at most exp x
- Reduction formula
- Rolle's theorem
- First derivative test
- cos pi = -1
- sin x / x tends to 1
- A function with nonnegative derivative is increasing
- Inverse trigonometric functions
- Cauchy mean value theorem
- Leibniz formula for pi
- Integral of exp(-x) over the half line
- Pythagorean trigonometric identity
- log x is at most x - 1
- Inverse hyperbolic functions
- Integral of cosine
- Derivative of arctangent
- Harmonic numbers grow like log n
- Integral of the exponential function
- A function with positive derivative is strictly increasing
- Derivative of the logarithm
- Integration by parts
- Arcsine
- Mean value theorem
- pi
- Uniform convergence of derivatives allows term-by-term differentiation
- Integration by substitution
- The exponential function is convex
- Basel problem
- Trigonometric substitution
- The gamma function extends the factorial
- Second derivative test
- Derivative of arcsine
- Area under a curve
- Integral of sine
- Dirichlet integral
- Taylor's theorem with the Lagrange remainder
- Logarithm of a quotient
- L'Hôpital's rule
- Integral of 1/x
- Logarithm of a product
- Stirling's formula
- The exponential function is strictly increasing
- Derivative of x^a
- Fundamental theorem of calculus
- Sine is 2π-periodic
- A function with bounded derivative is Lipschitz
- Fermat's theorem on stationary points
- Integral of a power
- Darboux's theorem
- Irrationality of pi
- pi is greater than 3
- Arccosine
- Logarithm is the inverse of the exponential
- Power-reduction formula for cosine
- Derivative of tangent
- Power series of arctangent
- Term-by-term differentiation
- Real exponentiation
- Derivative of cosine
- Logarithm of a power
- Improper integral of x^p on an unbounded interval
- Tangent half-angle substitution
- Integration by partial fractions
- x^2 sin(1/x) is differentiable with discontinuous derivative
- Taylor's theorem with the integral remainder
- Wallis product
- Second fundamental theorem of calculus
- Mercator series
- A function with nonnegative second derivative is convex
- The logarithm is strictly concave
- Arnold tongue
- Lebesgue's theorem on differentiability of monotone functions
- Minimal curvature property of natural cubic splines
- Stein's unbiased risk estimate
- Weierstrass product for the gamma function
- Stein's lemma
- Bertrand's test
- Mertens' theorems
- Peano kernel theorem
- Stein's example
- Reduction of an initial value problem to a Volterra equation
- Ramanujan's series for 1/pi
- Difference quotient
- Quadratic convergence of Newton's method
- Failure of Aizerman's conjecture
- Frullani integral
- Darboux function
- Gauss's test
- Log-convexity of the gamma function
- Mean value theorem for divided differences
- Differentiation operator as inverse of integration
- Logarithmically convex function
- Cauchy formula for repeated integration
- Interpolation error formula
- Borel's lemma
- Viète's formula
- Cantor function
- Error bound for the trapezoidal rule
- Bohr–Mollerup theorem
- Volterra's function
- An infinite product converges iff its series of logarithms converges
- Quintic equation
- Descartes' rule of signs
- Trigonometric integrals
- Basel problem via Fourier series
- Abel's summation formula
- Barnes G-function
- Chudnovsky algorithm
- Runge's phenomenon
Further reading
About this page
| Tag | 0NS8BP96QX |
|---|---|
| URI | m2://0NS8BP96QX |
| Aliases | relative-minimum |
| Kind | definition |
| Writing | W1 stated |
| MSC | 26A24 |
| Defines | Local minimum |
| Uses | Local maximum |
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What this page needs for W2
- edge instance_of in: one example (0 of 1)
- edge not_instance_of in: one non-example that fails one clause (0 of 1)
The writing levels say how complete the text of a page is. They do not measure its truth.