Lagrange Multipliers where the Constraint is an Inequality

Question

I'm currently learning about Lagrange multipliers in my multivariate calculus class and all of the problems we've been studying have had the constraint in the form of an equation $(g(x,y,z) = k)$ - however, I've also sometimes seen that problems involve both an equality constraint and an inequality constraint $(g(x,y,z) ≤ k)$ with the same expression. My intuition is that you can use Lagrange on the boundary (maximizing/minimizing $f(x,y,z)$ with constraint $g(x,y,z) = k$) and use partials for the interior (finding where $f(x,y,z)$ vanishes), but I'm not certain that this is the correct approach. Any input would be appreciated!

Answer 1

Yes, there is a generalisation of the Lagrange Multiplier theorem with regards to constraining inequalities by Karush-Kuhn-Tucker:

https://en.wikipedia.org/wiki/Karush%E2%80%93Kuhn%E2%80%93Tucker_conditions