I couldn't find a question answering this concept but they seem to be related.
Extreme Value Theorem (two variables)
If f is a continuous function defined on a closed and bounded set $A⊂\mathbb{R}^2$, then f attains an absolute maximum and absolute minimum value on A.
Lagrange Multipliers (two variables)
Extreme values of function f(x, y) subject to constraints g(x, y) = k has solutions in $\nabla f=\lambda \nabla g$.
The constraint in Lagrange Multipliers creates a closed and bounded region that would satisfy EVT, does it not? So does that make Lagrange multipliers a specific case of EVT?