Is there a way to apply the method of Lagrangian Multipliers when your equation of constraint is an inequality(it won't be called an equation of constraint then I guess). Here is an example of what I came up with.

Say

You want to find the extrema satisfying the inequality

If you introduce a variable that satisfies the equation that means

becomes your equation of constraint. Now we have

where

We get

From we get or

From we get or

For we get our minimum.

For we have and which is the local maximum.

Now what I want to know is, if there is a way to solve for more general inequalities like or .

Any input will be appreciated.