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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 Optimization Exercise 1
Imports
Step1:  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"
Step2:  Plot this function over the range $x\in\left[-3,3\right]$ with $b=1.0$ and $a=5.0$
Step3:  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.
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Python Code:
%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
import scipy.optimize as opt
Explanation: Optimization Exercise 1
Imports
End of explanation
# YOUR CODE HERE
def hat(x,a,b):
    v=-1*a*x**2+b*x**4
    return v
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
x=np.linspace(-3,3)
b=1.0
a=5.0
plt.plot(x,hat(x,a,b))
# YOUR CODE HERE
x0=-2
a = 5.0
b = 1.0
y=opt.minimize(hat,x0,(a,b))
y.x
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
# YOUR CODE HERE
x0=-2
a = 5.0
b = 1.0
i=0
y.x
mini=[]
x=np.linspace(-3,3)
for i in x:
    y=opt.minimize(hat,i,(a,b))
    z=int(y.x *100000)
    if np.any(mini[:] == z):
        i=i+1
    else: 
        mini=np.append(mini,z)
mini=mini/100000
mini
plt.plot(x,hat(x,a,b),label="Hat Function")
plt.plot(mini[0],hat(mini[0],a,b),'ro',label="Minima")
plt.plot(mini[1],hat(mini[1],a,b),'ro')
plt.xlabel=("X-Axis")
plt.ylabel=("Y-Axis")
plt.title("Graph of Function and its Local Minima")
plt.legend()
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
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