Finding global maxima and minima is harder. There is one case that is of interest.

We say that a domain is *convex* if every line drawn
between two points in the domain lies within the domain.

We say that
a function *f* is *convex* if the line connecting any two
points lies above the function. That is, for all *x*,*y* in the domain
and , we have , as before (see Chapter 2).

- If a function is convex on a convex domain, then any local minimum is a global minimum.
- If a function is concave on a convex domain, then any local maximum is a global maximum.

To check that a function is convex on a domain, check that its Hessian
matrix *H*(*x*)
is positive semidefinite for every point *x* in the domain. To check that
a function is concave, check that its Hessian is negative semidefinite
for every point in the domain.

The determinants of the principal minors are det ,
det and det . So
is positive semidefinite for all in .
This implies that *f* is convex over .

Mon Aug 24 16:30:59 EDT 1998