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MATH
[1251]

Definitions from Local extrema

All definitions from Local extrema.

Back to Local extrema

Local and absolute extrema

Let f:D→Rf:D\to\mathbb{R} and let x0∈D\mathbf{x}_0\in D.

  • ff has a local minimum at x0\mathbf{x}_0 if f(x)≥f(x0)f(\mathbf{x})\ge f(\mathbf{x}_0) for every x\mathbf{x} in DD sufficiently close to x0\mathbf{x}_0.
  • ff has a local maximum at x0\mathbf{x}_0 if f(x)≤f(x0)f(\mathbf{x})\le f(\mathbf{x}_0) for every x\mathbf{x} in DD sufficiently close to x0\mathbf{x}_0.
  • ff has an absolute, or global, minimum at x0\mathbf{x}_0 if f(x)≥f(x0)f(\mathbf{x})\ge f(\mathbf{x}_0) for every x∈D\mathbf{x}\in D.
  • ff has an absolute maximum at x0\mathbf{x}_0 if f(x)≤f(x0)f(\mathbf{x})\le f(\mathbf{x}_0) for every x∈D\mathbf{x}\in D.

A local classification cannot settle a global optimisation problem.

  1. [1]:Critical points depending on a parameter

Critical and stationary points

For f:D→Rf:D\to\mathbb{R}, an interior point x0∈int⁡(D)\mathbf{x}_0\in\operatorname{int}(D) is a critical point if ff is not differentiable at x0\mathbf{x}_0 or if ∇f(x0)=0\nabla f(\mathbf{x}_0)=\mathbf{0}. A point at which the gradient exists and is zero is called a stationary point.

Critical points include points where differentiability fails because such points can still be extrema. The stationary-point equations locate only the critical points with zero gradient; boundary points are handled separately in a global-extrema problem.

  1. [1]:The multivariable Fermat condition
  2. [2]:Seen Questions
  3. [3]:Seen Questions

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Local and absolute extremaCritical and stationary points