Optima can occur in three places:
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.
This function is everywhere differentiable, so extrema can only occur at points such that .
This equals 0 iff or (1,1). The Hessian is
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.