Maximum Using Lagrange

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NotTheBrightest

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Hello, I’m surely stuck on a task. It would be nice to receive some explanations for the task, but pure answer will do as well.

I need to find the maximum for z=15xy^2 on condition 5x+3y+30=480 using Lagrange’s function.

I have tried to look at some videos explaining this function, but it’s hard to understand without examples. Any help is appreciated ❤️
 
NotTheBrightest said:
Hello, I’m surely stuck on a task. It would be nice to receive some explanations for the task, but pure answer will do as well.

I need to find the maximum for z=15xy^2 on condition 5x+3y+30=480 using Lagrange’s function.

I have tried to look at some videos explaining this function, but it’s hard to understand without examples. Any help is appreciated ❤
Click to expand...
What is your textbook? Did you look into the example problems of the book?

Which videos did you see? Those did not show any examples?!
 
Subhotosh Khan said:
What is your textbook? Did you look into the example problems of the book?

Which videos did you see? Those did not show any examples?!
Click to expand...

Math for economists. I’m from western Europe so it won’t be the same for you. I don’t think it’s in my book as this was listed as a harder task, but I have to do it nonetheless.

The video had examples, but it was weird with using budget which I don’t have. Math is not my strongest side so excuse me if I sound dumb.
 
NotTheBrightest said:
Math for economists. I’m from western Europe so it won’t be the same for you. I don’t think it’s in my book as this was listed as a harder task, but I have to do it nonetheless.

The video had examples, but it was weird with using budget which I don’t have. Math is not my strongest side so excuse me if I sound dumb.
Click to expand...
"... I don’t think it’s in my book" - did you check thoroughly - using the "index" pages?

"The video had examples, but it was weird..." What were those videos?

Try:

"https://math.stackexchange.com/ques...ximum-of-a-function-using-lagrange-multiplier"

or

"https://services.math.duke.edu/~leili/teaching/duke/math212s16/lectures/lec10.pdf"
 
Subhotosh Khan said:
"... I don’t think it’s in my book" - did you check thoroughly - using the "index" pages?

"The video had examples, but it was weird..." What were those videos?

Try:

"https://math.stackexchange.com/ques...ximum-of-a-function-using-lagrange-multiplier"
Click to expand...
I’ll try to look again in the book. Could wolfram be used to help find max/min values in the future? I’m not sure if I’m allowed to post a link like this, but I can’t extract the youtube URL. This is the viseo I watched: https://www.khanacademy.org/math/mu...mization/v/lagrange-multiplier-example-part-1
Thank you for responding to me, I appreciate it.
 
Last edited:
[MATH] z=15xy^2 [/MATH][MATH]g = 5x+3y+30-480 = 5x+3y-450[/MATH]
We solve

[MATH]\nabla (z - \lambda g) = 0[/MATH]
i.e.

[MATH]15 y^2-5 \lambda = 0\\ 30 x y-3 \lambda=0\\ -5 x-3 y+450=0[/MATH]

There are two solutions

[MATH](x,y) = (90,0),~(30,100)[/MATH]
Evaluating \(\displaystyle z\) at these points we get

[MATH]z = (0,4500000)[/MATH]
So the first solution corresponds to a minimum, the second to a maximum.
 
Romsek said:
[MATH] z=15xy^2 [/MATH][MATH]g = 5x+3y+30-480 = 5x+3y-450[/MATH]
We solve

[MATH]\nabla (z - \lambda g) = 0[/MATH]
i.e.

[MATH]15 y^2-5 \lambda = 0\\ 30 x y-3 \lambda=0\\ -5 x-3 y+450=0[/MATH]

There are two solutions

[MATH](x,y) = (90,0),~(30,100)[/MATH]
Evaluating \(\displaystyle z\) at these points we get

[MATH]z = (0,4500000)[/MATH]
So the first solution corresponds to a minimum, the second to a maximum.
Click to expand...
Thanks, how did you get the (30,100)? I understand that (90,0) comes from 450/5, but then shouldn’t the other value be (150,0)? I’m probably missing something, but better safe than sorry.
 
You have to solve the for the gradient being zero.

Those 3 equations in post #6, just below "i.e."
 
NotTheBrightest said:
Hello, I’m surely stuck on a task. It would be nice to receive some explanations for the task, but pure answer will do as well.

I need to find the maximum for z=15xy^2 on condition 5x+3y+30=480 using Lagrange’s function.

I have tried to look at some videos explaining this function, but it’s hard to understand without examples. Any help is appreciated ❤
Click to expand...

Just to give you slightly different notation...we have the objective function (the function we are asked to omptimize):

[MATH]z=f(x,y)=15xy^2 [/MATH]
Subject to the constraint:

[MATH]g(x,y)=5x+3y-450=0[/MATH]
Using Lagrange Multipliers, we obtain the system:

[MATH]f_x(x,y)=15y^2=\lambda(5)=\lambda g_x(x,y)[/MATH]
[MATH]f_y(x,y)=30xy=\lambda(3)=\lambda g_y(x,y)[/MATH]
This system implies:

[MATH]\lambda=3y^2=10xy[/MATH]
[MATH]y(3y-10x)=0[/MATH]
From this we have two cases:

i) [MATH]y=0[/MATH]
Substitute into the constraint:

[MATH]5x+3(0)-450=0\implies x=90[/MATH]
ii) [MATH]y=\frac{10}{3}x[/MATH]
Substitute into the constraint:

[MATH]5x+3\left(\frac{10}{3}x\right)-450=0\implies x=30\implies y=100[/MATH]
And so we have the two critical points:

[MATH](x,y)\in\left\{(90,0),(30,100)\right\}[/MATH]
We find:

[MATH]f_{\min}=f(90,0)=0[/MATH]
[MATH]f_{\max}=f(30,100)=4500000[/MATH]
 
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