maximizing using lagrangian

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So I have a question from my quiz. I don't want a specific answer but a help or guidence. My objective function is F(x,y)=x+4y and my subject is I-Pxx-Pyy=0 where I,Px and Py are both positive integers. When I am trying to use lagrangian and try to maximize objective function I just can't but I don't know why. How can I maximize this function.

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Using lagrange multipliers and differentiating, we get $$\frac{\partial L}{\partial x}=1+\lambda P_x=0$$ $$\frac{\partial L}{\partial y}=5+\lambda P_y=0.$$ Unless $P_x$ and $P_y$ take on special values, this does not have a solution. This indicates we do not have an interior optimum and we must check the boundary conditions. This means checking if $x=0$ or $y=0$. Depending on prices, it will be one or the other, except in the case where the answer is indeterminant.