To derive Formula 3.11, we observe that the slope of the Capital Market Line connecting $R_0$ to $M$ is the maximum among all lines connecting $R_0$ to any feasible portfolio. The weights for MP are derived by solving the following: maximize the slope $(w^T\mu -R_0)/\sqrt{w^{T}Cw}$ subject to $w^{T}u=1$. Derive the formula by completing the following steps:

(a) Form the Lagrangian and using $\left(\partial /\partial w\right)w^{T}Cw=2Cw$ and the chain rule, show

$$\frac{\partial L}{\partial w}=\frac{\mu -\frac{Cw}{w^{T}Cw}(w^T\mu -R_0)}{\sqrt{w^{T}Cw}}-\lambda u$$

and set the Lagrangian to zero to get

$$\mu -\frac{w^T\mu -R_0}{w^{T}Cw}Cw=\left(\lambda \sqrt{w^{T}Cw}\right)u$$

(b) Find the vector $v$ so that when you pre-multiply both sides by $v^T$, you end up with

$$\begin{array}{c}\lambda =R_0/\sqrt{w^{T}Cw}\\ \mu -\frac{w^T\mu -R_0}{w^{T}Cw}Cw=R_{0}u\end{array}$$

(c) Find the matrix A so that when you pre-multiply both sides of the above by $A$, you end up with

$$\begin{array}{}\frac{w^T\mu -R_0}{w^{T}Cw}w=C^{-1}(\mu -R_{0}u)& \text{(3.26)}\end{array}$$

(d) Find the vector $v$ so that when you pre-multiply both sides of Formula 3.26 by