First-Order Equations · Superchallenge problem

A positive continuous function $f$ on $[0,2)$ has $f(0)=1$ and $f(1)=4$. For every $x$ in $(0,2)$, the average value of $f$ on $[0,x]$ equals the geometric mean $\sqrt{f(0)\,f(x)}$. Find $f\!\left(\dfrac32\right)$.