\[ f(x)=\ln (2 x+1)+x^{2}-5 \quad x>-0.5 \] - Show that \(f(x)=0\) has a root in the interval \([1.8,1.9]\)

Question

\[
f(x)=\ln (2 x+1)+x^{2}-5 \quad x>-0.5
\]
(a) Show that \(f(x)=0\) has a root in the interval \([1.8,1.9]\)
(b) Find \(\mathrm{f}^{\prime}(x)\)
(c) Using \(x_{0}=1.8\) as a first approximation, apply the Newton-Raphson procedure to find a second approximation, giving your answer to 3 decimal places.