(i) Suppose that $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$, and $g: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ are continuously differentiable. Suppose that the problem

$\max f(x)$ subject to $g(x)=b$

is solved by a unique $\bar{x}=\bar{x}(b)$ for each $b \in \mathbb{R}^{m}$, and that there exists a unique $\lambda(b) \in \mathbb{R}^{m}$ such that

$$\varphi(b) \equiv f(\bar{x}(b))=\sup _{x}\left\{f(x)+\lambda(b)^{T}(b-g(x))\right\}$$

Assuming that $\bar{x}$ and $\lambda$ are continuously differentiable, prove that

$$\frac{\partial \varphi}{\partial b_{i}}(b)=\lambda_{i}(b)$$

(ii) The output of a firm is a function of the capital $K$ deployed, and the amount $L$ of labour employed, given by

$$f(K, L)=K^{\alpha} L^{\beta}$$

where $\alpha, \beta \in(0,1)$. The firm's manager has to optimize the output subject to the budget constraint

$$K+w L=b,$$

where $w>0$ is the wage rate and $b>0$ is the available budget. By casting the problem in Lagrangian form, find the optimal solution and verify the relation $(*)$.