List of top Mathematics for Economy Questions on Optimization asked in IIT JAM EN

Let 𝑆 be a feasible set of a linear programming problem (𝑃). If the dual problem of (𝑃) is unbounded then

Consider a linear programming problem (𝑃) min 𝑧 = 4𝑥₁ + 6𝑥₂ + 6𝑥₃ subject to
𝑥₁+3𝑥₂≥3
𝑥₁+2𝑥₃ ≥5 
𝑥₁, 𝑥₂, 𝑥₃ ≥ 0 
If \(𝑥^∗ = (𝑥^∗_1 , 𝑥^∗_2 , 𝑥^∗_3 )\) is an optimal solution and 𝑧^∗ is an optimal value of (𝑃) and 𝑤^∗ =\((𝑤^∗_1 , 𝑤^∗_2 )\) is an optimal solution of the dual of (𝑃) then

For K∈R, let 𝑓(𝑥)=𝑥⁴+2𝑥³+𝑘𝑥²−𝑘, X∈R. If 𝑥=\(\frac{3}{2}\) is a point of local minima of 𝑓 and 𝑚 is the global minimum value of 𝑓 then 𝑓(0) − 𝑚 is equal to _______ (in integer).

If (𝑥^∗ , 𝑦^∗) is the optimal solution of the problem
maximize 𝑓(𝑥, 𝑦) = 100−𝑒^−𝑥−𝑒 ^−𝑦 
subject to 𝑒𝑥+𝑦=\(\frac{𝑒}{𝑒−1},\) 𝑥 ≥ 0, 𝑦 ≥ 0. 
Then \(\sqrt{\frac{y^*}{x^*}}\) is equal to ________ (round off to 2 decimal places).