# Math Help - Lagrange Multiplier

1. ## Lagrange Multiplier

Was going through a question that's already been done in the class. When I compared my results with my lecturer's I spotted I had $\lambda = 0$ for some critical points, whereas my lecturer only had non-zero values for it.

Is $\lambda$ always a non-zero value? I can't be sure as I have terrible notes.

2. I think that if lambda is 0 then your solution happens to lie on a stationary point of the (unconstrained) objective function, and so that particular constraint has no effect. Do you agree?

Spoiler:

suppose we're maximising f(x) subject to g(x) the lagrangian is $L = f(x) + \lambda g(x)$

at the optimal point:

$\dfrac{dL}{dx} = 0 = \dfrac{df}{dx} + \lambda$

so if lamda =0 then

$0 = \dfrac{df}{dx}$

and the unconstrained objective function is at a stationary point.

if there was more than one constraint, then the interpretation is a bit messier, of course.