I need help with these two questions
1.
Use Lagrange multipliers to show that, of all the triangles inscribed in a circle of radius
R, the equilateral triangle has the largest perimeter.
For this I know that the equation for the unit circle is
x^2+y^2 = 1
But i'm not sure what the constraint is.
Is it that for a triangle to be on a circle, each side must be <= to the diameter?
2.
In dimensions 2 and 3, the Lagrange condition can be replaced by a cross product condition. If a is an extremum for f(x) subject to a constraint g(x) = 0, then Gr[f] × Gr[g] = 0.
the x is a cross product sign
Gr is the gradient
fx is derivative of f w.r.t. x
If f and g are functions of 2 variables, Gr[f] × Gr[g] =
det
[e1 e2 e3]
[fx fy 0]
[gx gy 0]
= (fx gy − fy gx) e3.
Use the cross product condition to find the extrema of f(x, y) = x y on the unit circle.