CAPM from the tangency condition

(a) Show that maximising $S(w) = (w^\top\mu - r_f)/\sqrt{w^\top\Sigma w}$ over $w$ with $\mathbf 1^\top w = 1$ gives $\Sigma w = \kappa(\mu - r_f\mathbf 1)$ for some scalar $\kappa$; give $\kappa$ in terms of the optimal portfolio's mean and variance. (b) Using market clearing, derive (6.14). (c) Show that the CAPM implies that no portfolio has a higher Sharpe ratio than the market.