With usual notations, the properties of maxima and minima under various conditions are __________.

I		II
(P)	Maxima	(i)	r t− s2 = 0
(Q)	Minima	(ii)	rt − s2 < 0
(R)	Saddle Point	(iii)	rt − s2 > 0, r > 0
(S)	Case of failure	(iv)	rt − s2 > 0, r < 0

The correct answer is Option 1.

Key Points

• Maxima: The condition for a maximum point is given by the determinant condition rt − s² > 0 and r < 0.
• Minima: The condition for a minimum point is given by the determinant condition rt − s² > 0 and r > 0.
• Saddle Point: The condition for a saddle point is given by the determinant condition rt − s² < 0.
• Case of Failure: The case of failure is when the determinant condition rt − s² = 0.

Additional Information

• The second-order partial derivatives test is used to classify critical points as maxima, minima, or saddle points.
• The critical points are found by setting the first-order partial derivatives to zero and solving for the variables.
• The determinant of the Hessian matrix, given by D = rt − s², plays a crucial role in determining the nature of the critical points.
• Understanding these conditions is essential for solving optimization problems in calculus and applied mathematics.