2 marksMathematicsConstrained optimisation and linear programmingInequality constraints, second-order conditions and envelope theoremHardabout 3 min

IIT JAM 2021 Question 55: Constrained optimisation and linear programming

The optimal value of: minimize $2xy$ subject to $9x^2 + 4y^2 \le 36$ is _______ (in integer).

Answer and worked solution

Answer: -6

The minimum of $2xy$ occurs on the boundary $9x^2 + 4y^2 = 36$ with $x$ and $y$ of opposite sign. Lagrange conditions give $|y| = 1.5|x|$ , and the constraint yields $x^2 = 2$ .

Taking $x = \sqrt{2}$ , $y = -1.5\sqrt{2}$ gives $2xy = 2(\sqrt{2})(-1.5\sqrt{2}) = -6$ . The answer is $-6$ .

Plain-text version

The optimal value of: minimize 2xy subject to 9x² + 4y² ≤ 36 is _ (in integer).

Also written as: The optimal value of: minimize 2xy subject to 9x^2 + 4y^2 <= 36 is _ (in integer).

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.

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