Lagrangian Equation, Boundary Solution

Lagrangian Equation, Boundary Solution

I have a Lagrangian equation, which includes one multiplier with an aim to
maximise the total function (profit). At the first step, I tried a
positive multiplier (lambda>0), optimal solution for one of the decision
variables (production quantity) gives zero, which is impossible. Or, in
other word, I guess the profit function is convex rather than concave and
I guess I would get the boundary solution.
Can any body help how I can proceed to prove that my final solution is
indeed the boundary solution such that the KKT condtions do not produce
the optimal solution.