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

(I)		(II)
(P)	Maxima	(i)	rt − 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 4: (P)-(iv), (Q)-(iii), (R)-(ii), (S)-(i)

Concept:

In multivariable calculus, when analyzing a function f(x, y), we use the second-order partial derivatives to determine the nature of a critical point. With the usual notations:

• r = f_xx
• s = f_xy = f_yx
• t = f_yy
• D = rt − s² (the determinant of the Hessian matrix)

Interpretation of second-order conditions:

1. D = rt − s² = 0 → Case of failure: (P) = (i)
2. D = rt − s² > 0, r > 0 → Minima: (Q) = (iii)
3. D = rt − s² > 0, r < 0 → Maxima: (R) = (iv)
4. D = rt − s² < 0 → Saddle Point: (S) = (ii)

Now matching with the options:

• (P) Maxima → (i) rt − s² = 0 ❌
• (Q) Minima → (iii) rt − s² > 0, r > 0 ✅
• (R) Saddle Point → (iv) rt − s² > 0, r < 0 ❌
• (S) Case of failure → (ii) rt − s² < 0 ❌

Wait! Based on logic, the correct matching should be:

• Maxima → rt − s² > 0, r < 0 → (iv)
• Minima → rt − s² > 0, r > 0 → (iii)
• Saddle Point → rt − s² < 0 → (ii)
• Case of failure → rt − s² = 0 → (i)

So correct mapping is:

• (P) Maxima → (iv)
• (Q) Minima → (iii)
• (R) Saddle Point → (ii)
• (S) Case of failure → (i)

✅ This corresponds to: option 4, NOT option 1.

Hence, the correct answer is: option 4: (P)-(iv), (Q)-(iii), (R)-(ii), (S)-(i)