Answer
$$
3 \cos x=x+1
$$
The solutions of the given equation by Using Newton’s method, correct to six decimal places, are:
-3.637958, -1.862365, and 0.889470.
Work Step by Step
$$
3 \cos x=x+1
$$
We first rewrite the equation in standard form:
$$
3 \cos x-x-1=0
$$
Therefore we let
$$
f(x)=3 \cos x-x-1=0 .
$$
Then
$$
f^{'}(x)=-3 \sin x-1
$$
so, Formula 2 (Newton’s method) becomes
$$
\begin{aligned} x_{n+1} &=x_{n}-\frac{f(x)}{f^{'}(x)}\\
\\ &=x_{n}-\frac{3 \cos x_{n}-x_{n}-1}{-3 \sin x_{n}-1}\\
&=x_{n}+\frac{3 \cos x_{n}-x_{n}-1}{3 \sin x_{n}+1}\\
\end{aligned}
$$
In order to guess a suitable value for $x_1$ we sketch the graphs of $y=3 \cos x $ and $y = x+1$ in the Figure .
(*) It appears that they intersect at a point whose x-coordinate is somewhat less than 1, so let’s take $x_{1} = 1$ as a convenient first approximation. Then Newton’s method gives
$$
\begin{aligned} x_{2} &=x_{1}+\frac{3 \cos x_{1}-x_{1}-1}{3 \sin x_{1}+1}\\
&=1+\frac{3 \cos(1)-1-1}{3 \sin (1)+1} \approx 0.89243\dots
\end{aligned}
$$
repeating we get
$$
x_{3}\approx 0.889473,
x_{4}\approx 0.889470 \approx x_{5}
$$
(**) It appears that they intersect at a point whose x-coordinate is somewhat greater than -2, so let’s take $x_{1} = -2$ as a convenient first approximation. Then Newton’s method gives
$$
\begin{aligned} x_{2} &=x_{1}+\frac{3 \cos x_{1}-x_{1}-1}{3 \sin x_{1}+1}\\
&=-2+\frac{3 \cos(-2)-(-2)-1}{3 \sin (-2)+1} \approx -1.85621\dots
\end{aligned}
$$
repeating we get
$$
x_{3}\approx -1.862356,
x_{4}\approx -1.862365 \approx x_{5}
$$
(***) It appears that they intersect at a point whose x-coordinate is somewhat greater than -4, so let’s take $x_{1} = -4$ as a convenient first approximation. Then Newton’s method gives
$$
\begin{aligned} x_{2} &=x_{1}+\frac{3 \cos x_{1}-x_{1}-1}{3 \sin x_{1}+1}\\
&=-4+\frac{3 \cos(-4)-(-4)-1}{3 \sin (-4)+1} \approx -3.68228\dots
\end{aligned}
$$
repeating we get
$$
x_{3}\approx -3.638960 ,
x_{4} \approx - 3.637959 ,
x_{5} \approx -3.637958 \approx x_{6}
$$
We conclude that the roots of the equation, correct to six decimal places, are -3.637958, -1.862365, and 0.889470.