Conjugate gradient and Krylov optimality

For $f(x)=\tfrac12x^\top Ax - b^\top x$ with $A\succ0$:

1. Show that CG's $t$-th iterate minimises $f$ over the Krylov subspace $x_0 + \mathcal K_t = x_0 + \operatorname{span}\{r_0,Ar_0,\dots,A^{t-1}r_0\}$.
2. Deduce that CG terminates in at most $m$ steps, where $m$ is the number of distinct eigenvalues of $A$.
3. Explain why a clustered spectrum makes CG fast even when $\kappa$ is large.