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Questions › GATE › GATE 2025 › Q63
2 marksMathematics › Constrained optimisation and linear programming › Inequality constraints, second-order conditions and envelope theoremModerateabout 2 min

GATE 2025 Question 63: Constrained optimisation and linear programming

Let f(x,y)=−x2−y2+2x+4y+5f(x,y) = -x^2-y^2+2x+4y+5f(x,y)=−x2−y2+2x+4y+5.

Let (x∗,y∗)(x^*,y^*)(x∗,y∗) denote the solution to the following optimization problem:

Maximize f(x,y)subject to x≥0, y≥0, 2x+y≤6\text{Maximize } f(x,y) \quad \text{subject to } x\ge 0,\ y\ge 0,\ 2x+y\le 6Maximize f(x,y)subject to x≥0, y≥0, 2x+y≤6

Then the value of f(x∗,y∗)f(x^*,y^*)f(x∗,y∗) is ______ (in integer)

Answer and worked solution

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Plain-text version

Let f(x,y) = -x²-y²+2x+4y+5. Let (x*,y*) denote the solution to the following optimization problem: Maximize f(x,y) subject to x≥ 0, y≥ 0, 2x+y≤ 6 Then the value of f(x*,y*) is (in integer)

Also written as: Let f(x,y) = -x^2-y^2+2x+4y+5. Let (x*,y*) denote the solution to the following optimization problem: Maximize f(x,y) subject to x>= 0, y>= 0, 2x+y<= 6 Then the value of f(x*,y*) 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.

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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