Final answer:
The question involves finding and classifying critical points of a function in Mathematics, by analyzing first and second-order partial derivatives and the Hessian determinant.
Step-by-step explanation:
The subject of this question is Mathematics, specifically the classification of critical points for a given function of two variables f(x,y) = x³y³ - 9xy + 27. To determine the nature of each critical point, one must first find the first-order partial derivatives of f(x,y) with respect to x and y, set them equal to zero, and solve for x and y to find the critical points. After finding the critical points, the second-order partial derivatives are used to calculate the determinant of the Hessian matrix at each critical point to classify them as local minima, local maxima, or saddle points.
A critical point occurs when both first-order partial derivatives of the function are zero. To classify these points, the Hessian determinant at the critical point, given by D = f_xx(x,y)*f_yy(x,y) - (f_xy(x,y)) ^2 where subscripts denote partial derivatives, is used. If D > 0 and f_xx > 0, the point is a local minimum; if D > 0 and f_xx < 0, it's a local maximum. If D < 0, the point is a saddle point. If D = 0, the test is inconclusive