%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
import scipy.optimize as opt

"""
Explanation: Optimization Exercise 1
Imports
End of explanation
"""


def hat(x,a,b):
    return -a*x**2+b*x**4

assert hat(0.0, 1.0, 1.0)==0.0
assert hat(0.0, 1.0, 1.0)==0.0
assert hat(1.0, 10.0, 1.0)==-9.0

"""
Explanation: Hat potential
The following potential is often used in Physics and other fields to describe symmetry breaking and is often known as the "hat potential":
$$ V(x) = -a x^2 + b x^4 $$
Write a function hat(x,a,b) that returns the value of this function:
End of explanation
"""


a = 5.0
b = 1.0

X=np.linspace(-3,3,1000)
f=plt.figure(figsize=(15,10))
plt.plot(X,hat(X,a,b))

assert True # leave this to grade the plot

"""
Explanation: Plot this function over the range $x\in\left[-3,3\right]$ with $b=1.0$ and $a=5.0$:
End of explanation
"""


x1=opt.minimize(hat,3.0,args=(a,b))
x2=opt.minimize(hat,-3.0,args=(a,b))
print(x1.x, "\n")
print(x2.x)
f=plt.figure(figsize=(15,10))
plt.plot(X,hat(X,a,b))
plt.plot(x1.x,hat(x1.x,a,b), 'ro')
plt.plot(x2.x,hat(x2.x,a,b), 'ro')

assert True # leave this for grading the plot

"""
Explanation: Write code that finds the two local minima of this function for $b=1.0$ and $a=5.0$.

Use scipy.optimize.minimize to find the minima. You will have to think carefully about how to get this function to find both minima.
Print the x values of the minima.
Plot the function as a blue line.
On the same axes, show the minima as red circles.
Customize your visualization to make it beatiful and effective.
End of explanation
"""


