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.