Maxima and Minima of Functions of Two VariablesThe problem of determining the maximum or minimum of function is encountered in geometry, mechanics, physics, and other fields, and was one of the motivating factors in the development of the calculus in the seventeenth century. Let us recall the procedure for the case of a function of one variable y=f(x). First, we determine points x_c where f'(x)=0. These points are called critical points. At critical points the tangent line is horizontal. This is shown in the figure below.

The second derivative test is employed to determine if a critical point is a relative maximum or a relative minimum. If f''(x_c)>0, then x_c is a relative minimum. If f''(x_c)0 and f_xx(x_c,y_c)0 and f_xx(x_c,y_c)>0, then f(x,y) has a relative minimum at (x_c,y_c). If D0 with f_xx(1,0)=-2exp(2/3)