Find the maximum and minimum values of the function
f(x,y) = 2x^2 - 28xy + 2y^2 - 1 on the disk x^2 + y^2 <= 1.
(a) Find the maximum.
(b) Find the minimum.
Lagrange Multiplier method and partial differentiation.
The Attempt at a Solution
My work is attached as MyWork.jpg. If anything is unclear, tell me and I will restate it (or whatever it is you request). In the image, I forgot to plug in x and y in f(x,y) to get f(sqrt(1/2), sqrt(1/2). This yields the correct answer for the minimum: 2*(sqrt(1/2))^2 - 28*sqrt(1/2)*sqrt(1/2) + 2(sqrt(1/2))^2 - 1 whereas the correct answer for the maximum is: 2*(sqrt(1/2))^2 + 28*sqrt(1/2)*sqrt(1/2) + 2(sqrt(1/2))^2 - 1. So, my ultimate question is, what are the reasons for choosing positive or negative square roots for x and/or y?