# Plot the functions and the results
fig = plt.figure(figsize=(10, 5), dpi=300)

plt.plot(x, alphas[0] + betas[0] * np.log(x), color='red', label='Google Ads')
plt.plot(x, alphas[1] + betas[1] * np.log(x), color='blue', label='Facebook Ads')
plt.plot(x, alphas[2] + betas[2] * np.log(x), color='green', label='Twitter Ads')

# Plot optimal points
plt.scatter([google.value, facebook.value, twitter.value],
            [alphas[0] + betas[0] * np.log(google.value),
             alphas[1] + betas[1] * np.log(facebook.value),
             alphas[2] + betas[2] * np.log(twitter.value)],
            marker="+", color='black', zorder=10)

plt.xlabel('Budget ($)')
plt.ylabel('Returns ($)') 
plt.legend()
plt.show()