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Questions › UGC-NET › UGC-NET 2024 › Q69
2 marksMathematics › Constrained optimisation and linear programming › Inequality constraints, second-order conditions and envelope theoremHardabout 2 min

UGC-NET 2024 Question 69: Constrained optimisation and linear programming

Given the objective function y=f(x1,x2…xn)y=f(x_1, x_2 \ldots x_n)y=f(x1​,x2​…xn​) and the constraint g(x1,x2…xn)=0g(x_1, x_2 \ldots x_n) = 0g(x1​,x2​…xn​)=0, the second order condition for an extremum is :
A. ∣Hˉ2∣<0,∣Hˉ3∣<0|\bar{H}_2| < 0, |\bar{H}_3| < 0∣Hˉ2​∣<0,∣Hˉ3​∣<0 for a maximum
B. ∣Hˉ2∣>0,∣Hˉ3∣<0|\bar{H}_2| > 0, |\bar{H}_3| < 0∣Hˉ2​∣>0,∣Hˉ3​∣<0 for a maximum
C. ∣Hˉ2∣<0,∣Hˉ3∣>0|\bar{H}_2| < 0, |\bar{H}_3| > 0∣Hˉ2​∣<0,∣Hˉ3​∣>0 for a maximum
D. ∣Hˉ2∣<0,∣Hˉ3∣<0|\bar{H}_2| < 0, |\bar{H}_3| < 0∣Hˉ2​∣<0,∣Hˉ3​∣<0 for a minimum
E. ∣Hˉ2∣<0,∣Hˉ3∣<0|\bar{H}_2| < 0, |\bar{H}_3| < 0∣Hˉ2​∣<0,∣Hˉ3​∣<0 for a minimum Choose the correct answer from the options given below:

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Answer and worked solution

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Plain-text version

Given the objective function y=f(x₁, x₂ … xₙ) and the constraint g(x₁, x₂ … xₙ) = 0, the second order condition for an extremum is: A. |H̄₂| < 0, |H̄₃| < 0 for a maximum B. |H̄₂| > 0, |H̄₃| < 0 for a maximum C. |H̄₂| < 0, |H̄₃| > 0 for a maximum D. |H̄₂| < 0, |H̄₃| < 0 for a minimum E. |H̄₂| < 0, |H̄₃| < 0 for a minimum Choose the correct answer from the options given below: (A) A & D Only (B) C & E Only (C) B & D Only (D) B & E Only

Also written as: Given the objective function y=f(x_1, x_2... x_n) and the constraint g(x_1, x_2... x_n) = 0, the second order condition for an extremum is: A. |H_bar_2| < 0, |H_bar_3| < 0 for a maximum B. |H_bar_2| > 0, |H_bar_3| < 0 for a maximum C. |H_bar_2| < 0, |H_bar_3| > 0 for a maximum D. |H_bar_2| < 0, |H_bar_3| < 0 for a minimum E. |H_bar_2| < 0, |H_bar_3| < 0 for a minimum Choose the correct answer from the options given below:

Constrained optimisation and linear programming has come up 44 times in 7 exams since 2007.

What this question tests

This question is on Inequality constraints, second-order conditions and envelope theorem, part of Constrained optimisation and linear programming in Mathematics.

Optima with inequality constraints, tests for maxima, and how the optimal value changes with parameters.

TestStop has 8 past-paper questions on this subtopic; the nearest in year are below.

  • IIT JAM 2021 · Q55The optimal value of: minimize 2xy subject to 9x² + 4y² ≤ 36 is _ (in integer).
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