Lagrange Multipliers/Stationary Points with Constraint and Integration of x-y plane

[miss confuse]

Hello everyone!

Miss Confuse here, and I'm new in this forum. Would really appreciate anybody's help on these questions:
1. Use Lagrange multipliers to find the shortest distance from the origin of the hyperbola

x^2 + 8xy + 7y^2 = 225

2. Find all the stationary points of the function:

f(x,y) = (x+y)^4 -x^2 - y^2 -6xy

subject to constraint:

x^2 + y^2 = 4

3. Sketch the region of integration in the x-y plane for the following integral

I1: Integrate (upper limit = e, lower limit = 1) Integrate (upper limit = ln x, lower limit = 0) (y/x + e^-(2y+1) cos (xe^-(y+1)) dy dx

Change the order of integration showing clearly what the new limits of integration should be, and hence evaluate the integral.


What I've tried doing thus far:

1. I'm quite confuse with this one because I thought Lagrange multipliers require a constraint? As in f(x,y) - (lamda) g(x,y). Because of this, I don't really know what to do with this.

2. For this question, I did the following:
New Equation : (x+y)^4 -x^2 - y^2 -6xy - (lamda)(x^2 + y^2 -4)

Differentiated them base on x, y and lamda:

fx: 4(x+y)^3 - 2x - 6y -2(lamda)x = 0
fy: 4(x+y)^3 -2y - 6x - 2(lamda)y = 0
f(lamda): 2(lamda)x + 2(lamda)y - 4 = 0

From here on I'm a bit stuck as I don't know how to get their stationary points. The cubic equation is really confusing me >.<

3. For this question, I am totally blank, and do not know how to sketch the region or integrate this. Any help, or even guidance on how to start would be very much appreciated.

Thank you for reading this, I really do appreciate any help I can get. Thank you again!

 

[soroban]

Hello, miss confuse!

Welcome aboard!

1. Use Lagrange multipliers to find the shortest distance from the origin
to the hyperbola: \(\displaystyle x^2 + 8xy + 7y^2 \:=\: 225\)

We want to minimize the distance from the Origin to a point.
The hyperbola is the constraint.

\(\displaystyle f(x,y,\lambda) \;=\;\left(x^2+y^2\right)^{\frac{1}{2}} + \lambda(x^2+8xy + 7y^2 - 225)\)

2. Find all the stationary points of the function: \(\displaystyle f(x,y) \:=\: (x+y)^4 -x^2 - y^2 -6xy\)
subject to constraint: .\(\displaystyle x^2 + y^2 \:=\:4\)

I did the following:

New function: .\(\displaystyle f(x,y,\lambda) \;=\;(x+y)^4 -x^2 - y^2 -6xy + \lambda(x^2 + y^2 -4)\)

Differentiated them with respect to \(\displaystyle x, y\) and \(\displaystyle \lambda\):

\(\displaystyle [1]\;\;f_x \;=\;4(x+y)^3 - 2x - 6y +2\lambda x \:=\: 0\)

\(\displaystyle [2]\;\;f_y \;=\;4(x+y)^3 -2y - 6x + 2\lambda y \:=\:0\)

\(\displaystyle [3]\;\;f_{\lambda} \;=\;x^2 + y^2 - 4 \:=\:0\)

From here on I'm a bit stuck . . .

\(\displaystyle \begin{array}{cc}\text{Subtract [1] - [2]:} & -2x + 2y - 6y + 6x + 2x\lambda - 2y\lambda \:=\:0 \\ \\ & 4x - 4y + 2x\lambda - 2y\lambda \:=\:0 \\ \\ & 4(x-y) + 2\lambda(x-y) \:=\:0 \\ \\ & (x-y)(4+2\lambda) \:=\:0 \end{array}\)

\(\displaystyle \text{Hence: }\:\begin{Bmatrix}x - y \:=\:0 & \Rightarrow & x \:=\:y \\ 4+2\lambda \:=\:0 & \Rightarrow & \lambda \:=\:\text{-}2 \end{Bmatrix}\)

Got it?

 

[miss confuse]

Hey, yeah, thank you so much, I finally understood it. Really appreciate the help, thanks once again!