Given the objective function $$y=f(x_1, x_2, \ldots, x_n)$$​ and the constraint $$g(x_1, x_2, \ldots, x_n)=0$$​, the second order condition for an extremum is:
A. $$|\bar{H}_2| < 0, |\bar{H}_3| < 0, \ldots$$​, for a maximum
B. $$|\bar{H}_2| > 0, |\bar{H}_3| < 0, \ldots$$​, for a maximum
C. $$|\bar{H}_2| < 0, |\bar{H}_3| > 0, \ldots$$​, for a maximum
D. $$|\bar{H}_2| > 0, |\bar{H}_3| > 0, \ldots$$, for a maximum
E. $$|\bar{H}_2| < 0, |\bar{H}_3| < 0, \ldots$$, for a minimum
Choose the correct answer from the options given below:

Correct option is D

Note- Statement D and Statement E were exactly same in this Previous Year Question of (Jan 2025). For error free question, we have modified Statement D.

Correct Option: 4. B & E Only
Explanation:
The topic is Constrained Optimization using the Bordered Hessian. To determine if a stationary point is a maximum or minimum, we evaluate the signs of the principal minors of the Bordered Hessian matrix $$|\bar{H}|$$​.
Information Booster:

• Condition for Maximum: The relevant Bordered Hessian determinants must alternate in sign, starting with positive.
  
  • $$|\bar{H}_2| > 0$$​​
    
  • $$|\bar{H}_3| < 0$$
  • $$|\bar{H}_4| > 0, etc.$$​​
  • Matches Statement B.
• Condition for Minimum: The relevant Bordered Hessian determinants must all be of the same sign (specifically negative).
  
  • $$|\bar{H}_2| < 0$$​​
    
  • ​$$|\bar{H}_3| < 0$$​​
  • ​$$|\bar{H}_4| < 0, etc.$$​​
    
  • Matches Statement D.
    

Additional Knowledge:

• $$n$$​ variables, $$m$$ constraints: The checking of minors usually starts from$$|\bar{H}_{m+1}|$$​. Here, with 1 constraint, we check starting from $$|\bar{H}_2|$$​.
  
• First Order Condition: The first order condition requires that the first derivatives of the Lagrangian function with respect to all variables ($$x_i$$ and $$\lambda$$​) equal zero.
  
• Unconstrained vs. Constrained: In unconstrained optimization, we use a standard Hessian. In constrained optimization (like this question), we use the Bordered Hessian (bordered by the constraint derivatives).