Extrema

Local

Let be defined on a region containing the point

• is a local max of if for all domain points in a disk centered at
• is a local min of if for all domain points in a disk centered at

Global

Let be defined on a region containing the point

• is a global max of if for all in the domain of
• is a global min of if for all in the domain of

Theorem

On a Closed and Bounded Domain, any continous function attains a global minimum and maximum

Strategy for finding global extrema

Let , domain be closed and bounded.

1. Find all critical points of inside , and on the Boundary of
2. Evaluate at each critical point, as well as any endpoints of the boundary
3. The smallest value found is the global minimum; the largest is the global maximum

Critical Point

An extremum or saddle point. Given , a point in the domain of is called a Critical Point when

Second Derivative Test

This is for classification of critical points Suppose is a critical point of and has continuous second partial derivatives. Then we have:

• If and , then is a local minimum.
• If and , then is a local maximum.
• If , then has a saddle point at .
• If , the test is inconclusive.

General Test

More generally, if has a critical point at then:

• If all eigenvalues of are positive, is concave up in every direction from and so has a local minimum at .
• If all eigenvalues of are negative, is concave down in every direction from and so has a local maximum at .
• If some eigenvalues of are positive and some are negative, is concave up in some directions from and concave down in others, so has neither a local minimum nor maximum at .
• If all eigenvalues of are positive or zero, may have either a local minimum or neither at .
• If all eigenvalues of are negative or zero, may have either a local maximum or neither at .

Constraints of Equality

The Objective Function (the function containing the extrema) can be constrained by constraint function(s)

• For some simple problems you can just use substitution.
• For for complicated problems, the method of Lagrange Multipliers can be used to convert it into an unconstrained problem whose number of variables is the original number of variables plus the original number of equality constraints.

  Lagrange Multipliers

  • Extrema of subject to constraint satisfy
  • Extrema of subject to constraints satisfy
  • Extrema is a consequence of:
    • The Gradient of being normal to the level curve , everywhere
    • The gradient of along the boundary imposed by the constraint is also normal to the level curve , wherever the Parameterization of along the boundary (; single-variable calculus) would have a zero derivative, i.e. where there are possible Extrema. along the boundary is normal to the level curve at possible extrema because its parameterization is flat.
  • See here for a great video explanation

  Examples

  Example 1

  Find the points on surface that are closest to the origin. Let be the distance from a point to the origin Let be the proxy to the distance function that we will use to solve this problem, because its extrema exactly correspond to that of . is our objective function Let be our constraint function Goal: find points where and Find Gradients: Solve this system:

  4. Blah blah blah… You do a bunch of different cases and get the following points: You can plug each of these into and you’ll find and

  Link to original