(a) Sketch the increasing logarithm through \((1,0)\) and the downward parabola with zeros 0 and 3 and maximum \((1.5,2.25)\). There is no intersection for \(0<x\le1\), because the logarithm is non-positive and the parabola is positive. For \(1<x\le1.5\), \(\ln x\le\ln1.5<2\le3x-x^2\). For \(x>1.5\), the parabola strictly decreases and the logarithm strictly increases, so they cross exactly once.

(b) Let \(F(x)=3x-x^2-\ln x\). Then \[F(2)=2-\ln2\approx1.306853>0,\] \[F(2.8)=0.56-\ln2.8\approx-0.469619<0.\] Continuity and the sign change show a root between 2 and 2.8.

(c) Start with \(x_0=2.4\) and carry unrounded values between steps. The displayed iterates are to 4 decimal places:

The iterates settle at approximately 2.63231. Thus \[\boxed{x=2.63}\quad\text{(2 decimal places)}.\]