To bracket the roots, we can evaluate the function on a grid of points and find where it changes sign. In the code below, lower_bounds are the values of $x=a$ on this grid immediately before a sign change, so the root is bracketed by the tuple (a, a+dx) where $dx$ is the grid point spacing.
The Newton-Raphson method works best if the analytical form for the first derivative can be given:
$$
f'(x) = \cos\left(\frac{3}{x}\right) + \frac{3}{x}\sin\left(\frac{3}{x}\right).
$$
This function is defined in the lambda expression fp.
import numpy as np
from scipy.optimize import brentq, newton
f = lambda x: 0.2 + x*np.cos(3/x)
x,dx = np.linspace(-1, 1, 1000, retstep=True)
lower_bounds = x[np.sign(f(x[1:])) != np.sign(f(x[:-1]))]
# Brent's method
brent_roots = np.array([brentq(f, a, a+dx) for a in lower_bounds])
# Newton's method: requires first derivative of f(x)
fp = lambda x: np.cos(3/x) + 3/x * np.sin(3/x)
newton_roots = np.array([newton(f, a, fp) for a in lower_bounds])
print('Roots of f(x) = 1/5 + x.cos(3/x):')
print('{:11s} {:11s}'.format('Brent', 'Newton'))
r = np.vstack((brent_roots, newton_roots)).T
for br, nr in r:
print('{:11.8f} {:11.8f}'.format(br,nr))
Output:
Roots of f(x) = 1/5 + x.cos(3/x):
Brent Newton
-0.59333063 -0.59333063
-0.40858197 -0.40858197
-0.25181946 -0.25181946
-0.22939031 -0.22939031
0.29285950 0.29285950
0.35492010 0.35492010
0.67969870 0.67969870