Max/Min of x2−2xy+7y2 on Ellipse x2+4y2=1 w/ Lagrange Multiplier

[plexus0208]

Homework Statement

Use the Lagrange Multiplier method to find the maximum and minimum values of x² − 2xy + 7y² on the ellipse x² + 4y² = 1.

Homework Equations

Lagrange multiplier method

The Attempt at a Solution

L(x,y,z,λ) = x² − 2xy + 7y² - λ(x² + 4y² - 1)
Find L_x, L_y, L_λ
Then, solve for x and y?

 

[ideasrule]

Where did you get the L(x,y,z,λ)? You don't need the z in there because we're dealing with functions of x and y, not functions of x, y, and z.

Also, λ is a constant: you don't need to solve for Lλ. You don't need to solve for Lz either because there's no such variable as z. Just set Lx=0 and Ly=0 and solve. Don't forget to satisfy the initial constraint of x^2+4y^2=1!

 

[HallsofIvy]

Just an added note: some people are taught to find $L_\lambda$. Because if we are trying to extremize F(x) subject to the constraint G(x)= 0, we look at $L= F(x)+ \lambda G(x)$, $L_{\lamba}= G(x)= 0$ is just the constraint again.

 

[plexus0208]

@ideasrule: Sorry, ignore the "z."
@HallsofIvy: Thanks. And yes, that's how I learned it.

Also, once I find the critical points, do I just plug them back into the original function I want to extremize to see if it's a max or min?