If we need to use the method of Lagrange multipliers to find extreme values of a function $f(x, y)$ on a triangle-shaped region in $R ^2$ , how many times would we have to run the method? How many variables does each Lagrange function have?
What about a function f(x, y, z) and its extreme values over a cube-shaped region in $R^3$?
What i attempted
I know that all triangles have 3 sides, so i can assume that there should be 3 extreme values and Let the length of each side of the triangle be $x,y,z$ respectively. Hence im assuming there should be 3 constrains to the Lagrange multipliers. It would probably be an area of the triangle that we would need to maximize. Hence we would have to run the method 3 times and there will be 3 variables. The second part of the question should be of a similar concept.Im unsure about this though. Would appreciate any help. Thanks