Two$-$fund theorem. Let w$\_$i be the optimal portfolio in the Markowitz model under a target expected return $\backslash$mu i$,$ i $=1,2.$ With any given target expected return $\backslash$mu $,$ the corresponding optimal portfolio under the same Markowitz model can be replicated by a linear combination f w$\_1$ and w$\_2.$

$($a$)$ Prove the above statement for the Markowitz model without risk$-$free asset.

$($b$)$ Prove the above statement for the Markowitz model with risk$-$free asset.