Non-Linear Optimization - Query

Here is how you can implement the Lagrangian multiplier approach and solve
the first-order KKT conditions to obtain the solution for Paul Smith's
example:

? ? ? ?require(BB)

f2 <- function(x) {
f <- rep(NA, length(x))
f[1] <- 1 + 2 * x[1] * x[3] ?# x[3] is the Lagrangian multiplier
f[2] <- 1 + 2 * x[2] * x[3]
f[3] <- x[1]^2 + x[2]^2 - 1 ?# the equality constraint
f
}

dfsane(par=rep(0, 3), fn=f2)