State and prove the Contraction Mapping Theorem.

Find numbers $a$ and $b$, with $a<0<b$, such that the mapping $T: C[a, b] \rightarrow C[a, b]$ defined by

$$T(f)(x)=1+\int_{0}^{x} 3 t f(t) d t$$

is a contraction, in the sup norm on $C[a, b]$. Deduce that the differential equation

$$\frac{d y}{d x}=3 x y, \quad \text { with } y=1 \text { when } x=0,$$

has a unique solution in some interval containing 0 .