Hi everyone.

I'm not that familiar with English math terminology so I hope that you'll bear with me.

Currently, I'm trying to maximize a function with two constraints, but I got stuck because of one of my constraints. My first constraint has both the variables [tex]x[/tex] and [tex]y[/tex], but my second constraint only has the variabley. The reason why I'm confused by this is that when I proceed to solve the problem, I have no use for the Lagrange multiplier [tex]\lambda[/tex]. I can simply solve [tex]L_{\lambda_{1}} = 0[/tex] and [tex]L_{\lambda_{2}} = 0[/tex]. This will enough to yield my results (the [tex]x[/tex] and [tex]y[/tex] coordinates). It is frustrating me because I need to put it into words, what I am doing (in terms of using Lagrange multipliers) and why I apparently had to skip the [tex]\lambda[/tex] all together.

The function that I'm trying to maximize is as follows:

[tex]f(x,y) = -0,01x^2 + 395x + 100y[/tex]

My constraints are these:

[tex]2x + y = 44,000[/tex]

[tex]y = 20,000[/tex]

I know that the correct answer (through using other methods of optimization) is:

[tex]x = 12,000[/tex]

[tex]y = 20,000[/tex]

The way that I've proceeded to solve this problem is by putting the respective functions and constraints into a formula that was taught at school:

[tex]L(x, y, \lambda_{1}, \lambda_{2}) = -0,01x^2 + 395x + 100y - \lambda_{1} * (2x + y - 44,000) - \lambda_{2} * (y - 20,000)[/tex]

I then proceed by figuring out the partial differentials of [tex]L[/tex] with respect to [tex]x[/tex], [tex]y[/tex], [tex]\lambda_{1}[/tex] and [tex]\lambda_{2}[/tex] to ultimately isolate [tex]x[/tex] and [tex]y[/tex]. I am getting the correct results, but there's no need to solve [tex]L_x = 0[/tex] and [tex]L_y = 0[/tex]. This is what's confusing me and what I'm having a hard time putting into words.

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