In the Lagrange multiplier method, let us say we have a constraint $f(x) \le C$.
So we say setup the lagrange $L = ... -\lambda (C - f(x))$ where "..." contains the objective function.
My question is, let us say $\lambda$ in our optimal solution is really large. What does that tell us about the constraint?
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In the case of Dido's problem, you want to maximalize the integral $$I[y]=\int_{x_1}^{x_2}y \mathrm{d}x$$ with the condition $$J[y]=\int_{x_1}^{x_2}\sqrt{1+y'^2} \mathrm{d}x$$ So your Lagrange function will be $$L=y+\lambda \sqrt{1+y'^2}$$ To this problem, the solution will be a circle, with radius $|\lambda|$. So it can have a meaning.
It tells us nothing, really.
If the constraint is given by, say, $$ x^2+y^2-1=0 $$ you will get one value of $\lambda$. If you instead use $$ 2x^2+2y^2-2=0 $$ then you get a different value for $\lambda$. Continuing this way, you can make any non-zero value for $\lambda$ you want from the exact same constraint, just by expressing the constraint differently (as long as there is a non-zero value for $\lambda$ in the first place).