For Problem 6.20, plot the function \(f(x)\) between \(x=0\) and 4. Then provide a graphical interpretation why points close to \(x=2.2\) would be poorer initial guesses.

Problem 6.20

Use Newton-Raphson to find one solution to the polynomial equation \(f(x)=y\), where \(y=7\) and \(f(x)=x^{4}+3 x^{3}-15 x^{2}-19 x+30\). Start with \(x(0)=0\) and continue until (6.2.2) is satisfied with \(\varepsilon=0.001\).