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MATH
[1251]

Methods from Global extrema

All methods from Global extrema.

Back to Global extrema

Find absolute extrema on a closed bounded region

For a continuous ff on a nonempty closed and bounded region DD, use the following workflow, checking differentiability wherever a derivative is used:

  1. find all stationary points in the differentiable interior of DD and separately record any interior points where ff is not differentiable;
  2. restrict ff to each boundary piece;
  3. find critical points of every restricted function, including endpoints and corners;
  4. evaluate ff at every candidate;
  5. compare the resulting values.

Neither part of the search can be skipped: a boundary point can have a larger or smaller value than every interior critical point and a corner may not be detected by a one-variable derivative.

  1. [1]:The multivariable extreme-value theorem
  2. [2]:Analyse a boundary by parameterisation
  3. [3]:Extrema on a rectangle
  4. [4]:Extrema on the closed unit disk

Analyse a boundary by parameterisation

To analyse a differentiable boundary curve, parameterise it as

r(t)=(x(t),y(t))\mathbf{r}(t)=(x(t),y(t))

and optimise the one-variable function

ϕ(t)=f(x(t),y(t)).\phi(t)=f(x(t),y(t)).

Include all parameter endpoints when the boundary is a segment or a piecewise-smooth curve and include corners separately.

Parameterisation converts the boundary part of a two-variable problem into one-variable calculus. When the boundary is naturally a level curve g=0g=0, Lagrange multipliers solve the boundary problem without a parameterisation.

  1. [1]:Find absolute extrema on a closed bounded region
  2. [2]:Extrema on the closed unit disk

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Find absolute extrema on a closed bounded regionAnalyse a boundary by parameterisation