\(f:x\mapsto e^{3x}\) for \(x\in\mathbb R\), and \(g:x\mapsto2x^2+1\) for \(x\geq0\).

(i) Write down the range of \(g\).

(ii) Show that \(f^{-1}g(\sqrt{62})=\ln5\).

(iii) Solve \(f'(x)=6g''(x)\), giving your answer in the form \(\ln a\), where \(a\) is an integer.

(iv) Sketch the graph of \(y=g\) and the graph of \(y=g^{-1}\), showing the points where the graphs meet the coordinate axes.