I need to solve this problem, it's been 12 years since I graduated so I really don't remember much :) hopefully some1 can help
There is a factory which have 2 suppliers where X and Y represent the amount paid to each supplier The factory has a profit function as such $$\begin{align*}F(X,Y) &= 0.000002X^2Y + 0.000008X^2 + 180,000\\ S.T.\\ X + Y &\leq 100000\\ X &\geq 20000\end{align*}$$
Edit: So I did the Following (L will mark lambda)
$$0.000002X^2Y + 0.000008X^2 + 180,000 + \lambda_1(100000 - x - y) + \lambda_2(-20000 + X) = 0$$ I did all the math for partial derivatives and I still can't get the right answer
I also used solver and wolfram alpha to solve it, and both gave the correct answer which I didn't manage to obtain
Can some one help me with the solution, inc all the step?
Edit, I think I got it now, will appreciate remarks:
first I decided to change the lagrange, removing the second constraint $$\begin{align*} 0.000002X^2Y + 0.000008X^2 + 180,000 + \lambda_1(100000 - x - y) = 0\end{align*}$$
next I solve the derivate
$$\frac{\delta(X,Y)}{\delta(X)} =0.000004XY + 0.000016X - \lambda_1 = 0$$
$$\frac{\delta(X,Y)}{\delta(Y)} = 0.000002X^2 - \lambda_1 = 0$$
$$\frac{\delta(X,Y)}{\delta(\lambda_1)} = 100000 - X - Y = 0$$
next I solved those 3 equations and got that X is 66666.669 and Y is 33333.34
and those also apply to the second constraint so all is ok, I just don't understand why Ignoring the second constraint in lagrage allowed me to solve it so easily
keep in mind I graduated 12 years ago, and I didn't touch this stuff since then :) thanks