If we want to maximize $f(x)$ subject to two constraints, one which says that $x< c$ $c>0$, and another that says that $x\geq 0 $. Assume there are no problems with either $x=0, x>0$ or $\mu =0 ,\mu >0$, where $\mu$ is the multiplier on the second constraint.

Let's say that we take the case where $x> 0$, then $\mu =0$ (complementary slackness) and we find and expression $\lambda$, where $\lambda$ is the multiplier on the first constraint, **and the expression we get for** $\lambda$ does not depend on either $x$ or $\mu$, then are we guaranteed that $\lambda$ has the same value in the second case, where $x=0, \mu >0$, even if we cannot explicitly solve for $\lambda$ in this case? Also assuming that no other parameters change.

If necessary, assume some conditions of $f$ such as concavity, convexity, etc. Or ask if there are some properties I'm forgetting about.