[Python] how to set up the initial value for curve_fit to find the best optimizing, not just...

I am trying to fit a power-law function, and in order to find the best fit parameter. However, I find that if the initial guess of parameter is different, the "best fit" output is different. Unless I find the right initial guess, I can get the best optimizing, instead of local optimizing. Is there any way to find the **appropriate initial guess ** ????. My code is listed below. Please feel free make any input. Thanks!

import numpy as np
import pandas as pd
from scipy.optimize import curve_fit
import matplotlib.pyplot as plt
%matplotlib inline

# power law function
def func_powerlaw(x,a,b,c):
return a*(x**b)+c

test_X = [1.0,2,3,4,5,6,7,8,9,10]
test_Y =[3.0,1.5,1.2222222222222223,1.125,1.08,1.0555555555555556,1.0408163265306123,1.03125, 1.0246913580246915,1.02]

predict_Y = []
for x in test_X:
predict_Y.append(2*x**-2+1)

If I align with default initial guess, which p0 = [1,1,1]


popt, pcov = curve_fit(func_powerlaw, test_X[1:], test_Y[1:], maxfev=2000)


plt.figure(figsize=(10, 5))
plt.plot(test_X, func_powerlaw(test_X, *popt),'r',linewidth=4, label='fit: a=%.4f, b=%.4f, c=%.4f' % tuple(popt))
plt.plot(test_X[1:], test_Y[1:], '--bo')
plt.plot(test_X[1:], predict_Y[1:], '-b')
plt.legend()
plt.show()


The fit is like below, which is not the best fit.

If I change the initial guess to p0 = [0.5,0.5,0.5]


popt, pcov = curve_fit(func_powerlaw, test_X[1:], test_Y[1:], p0=np.asarray([0.5,0.5,0.5]), maxfev=2000)


I can get the best fit

---------------------Updated in 10.7.2018-------------------------------------------------------------------------------------------------------------------------

As I need to run thousands to even millions of Power Law function, using @James Phillips's method is too expensive. So what method is appropriate besides curve_fit? such as sklearn, np.linalg.lstsq etc.

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