Local minima and maxima of multivariable functions

Local minima and maxima of multivariable functions

Lessons

Notes:

Definition of Critical Points

Recall for 1 variable functions, a critical point occurs at a point x0x_0 if f(x0)=0f'(x_0)=0 or f(x0)f'(x_0) does not exist. The concept is the same for 2 variable functions, except we must modify a few things.

The point (x0,y0)(x_0, y_0) is a critical point of f(x,y)f(x,y) if one of the following is true:

  1. fx(x0,y0)=0f_x(x_0,y_0)=0, fy(x0,y0)=0f_y(x_0,y_0)=0
  2. fx(x0,y0)f_x(x_0,y_0) and/or fy(x0,y0)f_y(x_0,y_0) does not exist

Types of Critical Points

There are 3 types of critical points:

  1. Local Maximum: occurs when f(x0,y0)f(x,y)f(x_0,y_0) \geq f(x,y) for all points (x,y)(x,y) that is around (x0,y0)(x_0, y_0). In other words, it's the biggest value of the function around it's region.
  2. Local Minimum: occurs when f(x0,y0)f(x,y)f(x_0,y_0) \leq f(x,y) for all points (x,y)(x,y) that is around (x0,y0)(x_0, y_0). In other words, it's the smallest value of the function around it's region.
  3. Saddle point: neither a local minimum or local maximum.

Classifying Critical Points
Suppose that (x0,y0)(x_0,y_0) is a critical point of f(x,y)f(x,y). To see whether it's a local maximum, or local minimum, or saddle point, we compute the following:

D=fxx(x0,y0)fyy(x0,y0)[fxy(x0,y0)]2D=f_{xx}(x_0,y_0)\cdot f_{yy}(x_0,y_0)-[f_{xy} (x_0,y_0)]^2

If:

  1. D>0D\gt0 and fxx(x0,y0)>0f_{xx} (x_0,y_0)\gt0 , then it is a local minimum
  2. D>0D\gt0 and fxx(x0,y0)<0f_{xx}(x_0,y_0)\lt0, then it is a local maximum
  3. D<0D\lt0, then it is a saddle point
  4. D=0D=0, then it could be any of the 3 types. Need to use other techniques to classify it.
  • Introduction
    Local Minima & Maxima of Multivariable Functions Overview:
    a)
    Definition of Critical Points
    • Critical points for 1 variable
    • Critical points for 2 variables
    • An example

    b)
    Types of Critical Points
    • Local maximum
    • Local minimum
    • Saddle point

    c)
    Classifying Critical Points
    • Calculate DD
    • D>0D\gt0 and fxx(x0,y0)>0f_{xx}(x_0,y_0)\gt0 \to local minimum
    • D>0D\gt0 and fxx(x0,y0)<0f_{xx}(x_0,y_0)\lt0 \to local maximum
    • D<0D\lt0 \to saddle point
    • D<0D\lt0 \to saddle point
    • D=0D=0 \to failed to classify
    • An example