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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
def hat(x,a,b):
    return -1*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(-3,3,100)
v = hat(x,a,b)
graph = plt.plot(x,v)
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
f = lambda g: hat(g,a,b)
x1 = float(opt.minimize(f,-2 ).x)
x2 = float(opt.minimize(f,2 ).x)
print(x1)
print(x2)
graph = plt.plot(x,v)
plt.plot([x1,x2],[f(x1),f(x2)],'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
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