5. A common use for Lagrange multipliers is to maximize prot subject to a budget constraint. How can a company make the most moneyr with what they currently have to spend? In that context, the multiplier 3. has a very concrete interpretation: 2. is the marginal maximum prot with respect to the constraint. More precisely, if M is the maximum prot obtained with budget Sit: and J. is the corresponding multiplier from the Lagrange multiplier system, the (SiL; = A. We will investigate this surprising fact. Let Lg :R" Ir ill be C1. Suppose f attains a unique maximum on S, = {x e R\" : g(x} = c} for each c E R. Dene M(c) = max{f[x} E R : x E El",g(x} = c}. Let it"{c} be the location of the maximum in Sc and let l*[c) be the multiplier in the Lagrange system corresponding to it"(c). You may assume x" and it\" are C1 functions in c. (5b) Prove that for a xed C E R, (x*[C),l*[C)) is a critical point of Mat, 2., C)