import numpy as np
from scipy import optimize
import matplotlib.pyplot as plt

x = np.arange(-10, 10, 0.1)
def f(x):
    return x**2 + 10*np.sin(x)


grid = (-10, 10, 0.1)
xmin_global = optimize.brute(f, (grid,))
xmin_local = optimize.fminbound(f, 0, 10)
root = optimize.fsolve(f, 1)  # our initial guess is 1
root2 = optimize.fsolve(f, -2.5)

xdata = np.linspace(-10, 10, num=20)
np.random.seed(1234)
ydata = f(xdata) + np.random.randn(xdata.size)

def f2(x, a, b):
    return a*x**2 + b*np.sin(x)

guess = [2, 2]
params, params_covariance = optimize.curve_fit(f2, xdata, ydata, guess)


fig = plt.figure()
ax = fig.add_subplot(111)
ax.plot(x, f(x), 'b-', label="f(x)")
ax.plot(x, f2(x, *params), 'r--', label="Curve fit result")
xmins = np.array([xmin_global[0], xmin_local])
ax.plot(xmins, f(xmins), 'go', label="Minima")
roots = np.array([root, root2])
ax.plot(roots, f(roots), 'kv', label="Roots")
ax.legend()
ax.set_xlabel('x')
ax.set_ylabel('f(x)')