Distribution of the largest and smallest uniforms

Let $U_1,\dots,U_n$ be independent and uniform on $[0,1]$. For both $M=\max_i U_i$ and $m=\min_i U_i$, find the cumulative distribution function, the density and the expected value.

Max: $F(x)=x^n$, $f(x)=nx^{n-1}$, mean $\tfrac{n}{n+1}$. Min: $F(x)=1-(1-x)^n$, $f(x)=n(1-x)^{n-1}$, mean $\tfrac{1}{n+1}$. The max is below $x$ only if every $U_i$ is, and the min is above $x$ only if every $U_i$ is; differentiate and integrate to get the density and mean.