Expand/Contract All

Maxima and Minima of Function of Two Variables

 
 

Taylor's Therem for Functions of Two Variables

Recall the Taylor expansion for a function of a single variable ~x, about the point ~x = ~a :

~{ _ f} ( ~a + ~h ) _ _ = _ _ ~f ( ~a ) _ + _ ~h fract{d ~f,d~x}( ~a ) _ + _ fract{~h^2,2#!} fract{d^2~f,d~x^2}( ~a ) _ + _ ...

The equivalent expansion for a function of two variables is given by:

~{ _ f} ( ~a + ~h , ~b + ~k ) _ _ = _ _ ~f ( ~a , ~b ) _ + _ rndb{~h fract{∂~f,∂~x} _ + _ ~k fract{∂~f,∂~y}} ( ~a , ~b )

_ _ _ _ _ _ _ _ + _ fract{1,2#!}rndb{~h^2 fract{∂^2~f,∂~x^2} _ + _ ~k^2 fract{∂^2~f,∂~y^2} _ + _ 2~h~k fract{∂^2~f,∂~x∂~y}} ( ~a , ~b ) _ + _ ...

or more succinctly:

~{f} ( ~a + ~h , ~b + ~k ) _ = _ ~f + ( ~h ~f_~x + ~k ~f_~y ) + ( ~h^2 ~f_{~x~x} + ~k^2 ~f_{~y~y} + 2~h~k ~f_{~x~y} ) ./ 2#! + ...

[All functions on the right hand side being evaluated at _ ( ~a , ~b ).]

Maxima and Minima of Functions of Two Variables

If _ ~f ( ~x , ~y ) _ has a maximum or minimum at point ( ~a , ~b ) [ i.e. _ ( ~a , ~b ) _ is a point of inflection ], then

~f_~x = 0 _ and _ ~f_~y = 0 , _ _ at ( ~a , ~b )

Now put _ &delta.~f ( ~a , ~b ) _ = _ ~{f} ( ~a + ~h , ~b + ~k ) - ~{f} ( ~a , ~b )

So at a maximum or minimum

&delta.~f _ ~~ _ ~h^2 ~f_{~x~x} + ~k^2 ~f_{~y~y} + 2~h~k ~f_{~x~y}

So ~f has a maximum at ( ~a , ~b ) if _ ~f_~x = 0 _ and _ ~f_~y = 0 , _ and _ &delta.~f < 0 for all small ~h and ~k. Similarly it has a maximum if _ &delta.~f > 0 for all small ~h and ~k. In particular _ ~f_~{xx} < 0 _ and _ ~f_~{yy} < 0 _ for a maximum ( putting ~h = 0 and ~k = 0 respectively) and _ ~f_~{xx} > 0 _ and _ ~f_~{yy} > 0 _ for a minimum

The approximation for &delta.~f can be written:

&delta.~f _ ~~ _ ( ( ~h ~f_~{xx} + ~k ~f_~{xy} )^2 - ~k^2 ( ~f_~{xy}^2 - ~f_~{xx} ~f_~{yy} ) ) ./ 2 ~f_~{xx} , _ _ providing _ ~f_~{xx} != 0.

or _ _ _ _ &delta.~f _ ~~ _ ( &Theta.^2 - ~k^2 &Delta. ) / 2 ~f_~{xx}, _ where _ &Theta. = ( ~h ~f_~{xx} + ~k ~f_~{xy} ) , _ and _ &Delta. = ( ~f_~{xy}^2 - ~f_~{xx} ~f_~{yy} ) .

Note that _ ( &Theta.^2 - ~k^2 &Delta. ) _ is positive if &Delta. is zero or negative, and that &Delta. is not dependant on the values of ~h and ~k. So for a point of inflection we have:

  • &Delta. =< 0 _ and _ ~f_~{xx} > 0 _ => _ &delta.~f > 0 _ => _ minimum.
  • &Delta. =< 0 _ and _ ~f_~{xx} < 0 _ => _ &delta.~f < 0 _ => _ maximum.
  • &Delta. > 0 _ then the value of &delta.~f is dependant on ~h and ~k, so this is a saddle-point.

What if ~f_~{xx} = 0 ? _ Note that the original approximation for &delta.~f is symmetic in ~x, ~y, ~h and ~k. So we can write

&delta.~f _ ~~ _ ( Λ^2 - ~h^2 &Delta. ) / 2 ~f_~{yy}, _ where _ Λ = ( ~k ~f_~{yy} + ~h ~f_~{xy} ) , _ &Delta. same as above.

and note that if _ ~f_~{xx} = 0 _ then &Delta. is positive, so we will have a saddle-point. A completely symmetric argument holds for ~f_~{yy} = 0.