Optima can occur in three places:
with 
.
We will identify the first type of point with Kuhn-Tucker conditions (see next chapter). The second type is found only by ad hoc methods. The third type of point can be found by solving the gradient equations.
In the remainder of this chapter, we discuss the important case
where .
To identify if a point with zero gradient is a local maximum or local
minimum, check the Hessian.
is positive definite then
is a local minimum. is negative definite, then
is a local maximum.
This function is everywhere differentiable, so extrema can only
occur at points such that .
This equals 0 iff 
or (1,1). The Hessian is
So,
Let 
denote the first principal minor of H(0,0) and
let 
denote its second principal minor (see Section 1.6).
Then det 
and det 
. Therefore
H(0,0) is neither positive nor negative definite.
Its first principal minor has
det 
and its second principal minor
has det 
. Therefore
H(1,1) is positive definite, which implies that
(1,1) is a local minimum.
The revenue from sales is 
.
The production costs are 
,
the development cost is $20,000 and the cost of advertizing is a.
Therefore, Jane and Jim's profit is
To find the maximum profit, we compute the partial derivatives of f and set them to 0:
Solving this system of two equations yields
We verify that this is a maximum by computing the Hessian.
det 
and det 
at the point p=63.25, a= 15,006.25. So, indeed, this solution
maximizes profit.
Setting these partial derivatives to 0 yields the unique solution
. The Hessian matrix is
The determinants of the principal minors are det ,
det 
and det 
. So H(0,0,0)
is positive definite and the solution is a minimum.
