I'm a bit confused about Lagrange multipliers. I know it works wonders if I only have equality constraints. Whenever I have inequality constraints, or both, I use Kuhn-Tucker conditions and it does the job.
But my question is, can I solve a inequality constraint problem using only Lagrange multiplier?
min $f(x)=x^4$ with $x \leq -1$.
$L(x,\lambda)=x^4+ \lambda(x+1)$, gives:
$4x^3+ \lambda =0$ and $x+1=0 \Leftrightarrow x=-1 , \lambda =4$
Is this correct? I just assumed equality.. What happens if the constraint is not active?